Hi,
pls try below xsl (modified your xsl):
<xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform" version="1.0">
	<xsl:output method="xml" indent="yes"/>
	<xsl:template match="Root">
		<ArticleSet>
			<Article>
				<Journal>
					<PublisherName>
						<xsl:apply-templates select="Source/PublisherName"/>
					</PublisherName>
					<JournalTitle>
						<xsl:apply-templates select="Source/Publication"/>
					</JournalTitle>
					<Issn>
						<xsl:apply-templates select="Source/Issn"/>
					</Issn>
					<Volume>
						<xsl:apply-templates select="Source/Volume"/>
					</Volume>
					<Issue>
						<xsl:apply-templates select="Source/Issue"/>
					</Issue>
					<PubDate>
						<Year>
							<xsl:apply-templates select="Source/Date/Year"/>
						</Year>
						<Month>
							<xsl:apply-templates select="Source/Date/Month"/>
						</Month>
					</PubDate>
				</Journal>
				<ArticleTitle>
					<xsl:apply-templates select="Story/articleTitle"/>.
                   <xsl:apply-templates select="Story/articleSubTitle"/>
				</ArticleTitle>
				<FirstPage>
					<xsl:apply-templates select="Source/FirstPage"/>
				</FirstPage>
				<LastPage>
					<xsl:apply-templates select="Source/LastPage"/>
				</LastPage>
				<!-- This is the section were I want <author> to be <Author> -->
				<AuthorList>
					<xsl:apply-templates select="Story/author"/>
				</AuthorList>
				<Abstract>
					<xsl:apply-templates select="Story/abstract_body"/>
				</Abstract>
			</Article>
		</ArticleSet>
	</xsl:template>
	<xsl:template match="Story/author">
		<Author>
			<xsl:copy-of select="./*"/>
		</Author>
	</xsl:template>
</xsl:stylesheet>

but i found your xsl file contains very less <xsl:template> elements,
you included a major chunk within a single <xsl:template> element. it
is recommended to define <xsl:template> element matching certain sets
of elements, child elements.