<?xml version="1.0" encoding="us-ascii"?><InstanceReport xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><Version>2.4.0.8</Version><ReportLongName>2111100 - Disclosure - Stock-Based Compensation Expense</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption /><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode /><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol /><contextRef><ContextID>D2014Q1YTD</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0001489096</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2013-04-01T00:00:00</PeriodStartDate><PeriodEndDate>2013-06-30T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS /><CurrencyCode /><OriginalCurrencyCode /></MCU><CurrencySymbol /><Labels><Label Key="CalendarSupplement" Id="0" Label="3 Months Ended" /><Label Key="Calendar" Id="1" Label="Jun. 30, 2013" /></Labels></Column></Columns><Rows><Row FlagID="0"><Id>1</Id><IsAbstractGroupTitle>true</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>1</Level><ElementName>us-gaap_DisclosureOfCompensationRelatedCostsSharebasedPaymentsAbstract</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:stringItemType</ElementDataType><SimpleDataType>string</SimpleDataType><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Disclosure of Compensation Related Costs, Share-based Payments [Abstract]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>us-gaap_DisclosureOfCompensationRelatedCostsShareBasedPaymentsTextBlock</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2014Q1YTD" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText>&lt;div style="font-family:Times New Roman;font-size:10pt;"&gt;&lt;div style="line-height:120%;padding-top:12px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;font-weight:bold;"&gt;Stock-Based Compensation Expense&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:48px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;Since the completion of the CHS Transactions on April&amp;#160;30, 2010, the board of directors has adopted and the shareholders have approved &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;two&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; stock option award plans.&amp;#160; The 2010 Thermon Group Holdings,&amp;#160;Inc. Restricted Stock and Stock Option Plan (&amp;#8220;2010 Plan&amp;#8221;) was approved on July&amp;#160;28, 2010.&amp;#160; The 2010 Plan authorized the issuance of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;2,767,171&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; stock options or restricted shares (on a post stock split basis).&amp;#160; On April&amp;#160;8, 2011, the board of directors approved the Thermon Group Holdings,&amp;#160;Inc. 2011 Long-Term Incentive Plan (&amp;#8220;2011 LTIP&amp;#8221;). The 2011 LTIP made available &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;2,893,341&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; shares of the Company&amp;#8217;s common stock that may be awarded to employees, directors or non-employee contractor's compensation in the form of stock options or restricted stock awards.&amp;#160;&lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:48px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;At &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;June&amp;#160;30, 2013&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;, there were &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;1,046,700&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; options outstanding.&amp;#160; Stock compensation expense was &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$366&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; and &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$58&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; during the three months ended &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;June&amp;#160;30, 2013&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; and &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;2012&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;, respectively.&amp;#160;  &lt;/font&gt;&lt;/div&gt;&lt;div style="line-height:120%;padding-top:12px;text-indent:48px;font-size:10pt;"&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; During the three months ended June 30, 2013, &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;6,000&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; restricted stock units were issued to our employees with a total fair value of &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$126&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;. The fair value was determined based on the closing price of our stock on the date of issuance which ranged from &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$20.70&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; to &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;$22.02&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt;. The restricted stock units will be expensed on a straight-line basis over a &lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;color:#000000;text-decoration:none;"&gt;five&lt;/font&gt;&lt;font style="font-family:inherit;font-size:10pt;"&gt; year period and will vest on each anniversary of the issuance date over the same period.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;</NonNumbericText><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>nonnum:textBlockItemType</ElementDataType><SimpleDataType>na</SimpleDataType><ElementDefenition>The entire disclosure for compensation-related costs for equity-based compensation, which may include disclosure of policies, compensation plan details, allocation of equity compensation, incentive distributions, equity-based arrangements to obtain goods and services, deferred compensation arrangements, employee stock ownership plan details and employee stock purchase plan details.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 718

 -SubTopic 10

 -Section 50

 -Paragraph 1

 -URI http://asc.fasb.org/extlink&amp;oid=6415400&amp;loc=d3e5047-113901



Reference 2: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 718

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -URI http://asc.fasb.org/extlink&amp;oid=6415400&amp;loc=d3e5070-113901



Reference 3: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 505

 -SubTopic 50

 -Section 50

 -Paragraph 1

 -URI http://asc.fasb.org/extlink&amp;oid=6406099&amp;loc=d3e25284-112666



Reference 4: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 123R

 -Paragraph 64, 65, A240

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



Reference 5: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 718

 -SubTopic 40

 -Section 50

 -Paragraph 1

 -URI http://asc.fasb.org/extlink&amp;oid=6418621&amp;loc=d3e17540-113929



Reference 6: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 718

 -SubTopic 10

 -Section 50

 -Paragraph 4

 -URI http://asc.fasb.org/extlink&amp;oid=6415400&amp;loc=d3e5444-113901



Reference 7: http://www.xbrl.org/2003/role/presentationRef

 -Publisher SEC

 -Name Staff Accounting Bulletin (SAB)

 -Number Topic 14



Reference 8: http://www.xbrl.org/2003/role/presentationRef

 -Publisher AICPA

 -Name Statement of Position (SOP)

 -Number 93-6

 -Paragraph 53

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Stock-Based Compensation Expense</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Stock-Based Compensation Expense</ReportName><MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel><SharesRoundingLevel>UnKnown</SharesRoundingLevel><PerShareRoundingLevel>UnKnown</PerShareRoundingLevel><ExchangeRateRoundingLevel>UnKnown</ExchangeRateRoundingLevel><HasCustomUnits>true</HasCustomUnits><IsEmbedReport>false</IsEmbedReport><IsMultiCurrency>false</IsMultiCurrency><ReportType>Sheet</ReportType><RoleURI>http://www.thermon.com/role/StockBasedCompensationExpense</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>2</NumberOfRows></InstanceReport>
