Why is my XSLT here stripping HTML tags

That's what xsl:value-of is defined to do. If you want to retain the tags, use xsl:copy-of.

Try it this way:

<xsl:stylesheet version="1.0"xmlns:xsl="http://www.w3.org/1999/XSL/Transform" ><xsl:output method="xml" encoding="utf-8" omit-xml-declaration="yes" /><xsl:template match="/root">    <xsl:text>{&#10;"text": "</xsl:text>    <xsl:apply-templates select="text/text()"/>    <xsl:text>"&#10;"continue": "</xsl:text>    <xsl:apply-templates select="continue/*"/>    <xsl:text>"&#10;}</xsl:text></xsl:template><xsl:template match="*">    <xsl:copy>        <xsl:apply-templates/>    </xsl:copy></xsl:template><xsl:template match="text()"><xsl:variable name="escaped-text">    <xsl:call-template name="replace-string">        <xsl:with-param name="text" select="."/>        <xsl:with-param name="replace" select="'"'" />        <xsl:with-param name="with" select="'\"'"/>    </xsl:call-template></xsl:variable><xsl:value-of select="normalize-space($escaped-text)"/></xsl:template><xsl:template name="replace-string">    <xsl:param name="text"/>    <xsl:param name="replace"/>    <xsl:param name="with"/>    <xsl:choose>        <xsl:when test="contains($text,$replace)">            <xsl:value-of select="substring-before($text,$replace)"/>            <xsl:value-of select="$with"/>            <xsl:call-template name="replace-string">                <xsl:with-param name="text"                    select="substring-after($text,$replace)"/>                <xsl:with-param name="replace" select="$replace"/>                <xsl:with-param name="with" select="$with"/>            </xsl:call-template>        </xsl:when>        <xsl:otherwise>            <xsl:value-of select="$text"/>        </xsl:otherwise>    </xsl:choose></xsl:template></xsl:stylesheet>

Applied to a modified version of your input (added root element and some more markup for testing):

<root>    <text>    Kevin Love and Steph Curry can talk about their first-    time starting gigs in the All-Star game Friday night when the Minnesota    Timberwolves visit Oracle Arena to face the Golden State Warriors.    </text>    <continue>        <P>        Love and Curry were <i>two of <b>four</b> first-time All-Star</i> starters when the league        made the announcement on Thursday.        </P>        <P>        Love got a late push to overtake Houston Rockets center Dwight Howard in the        final week of voting.        </P>        <P>        "I think it's a little sweeter this way because I really didn't expect it,"        Love said on a conference call. "I was already humbled by the response the        fans gave me to being very close to the top (frontcourt players). The outreach        by the Minnesota fans and beyond was truly amazing."        </P>    </continue></root>

produces the following result:

{"text": "Kevin Love and Steph Curry can talk about their first- time starting gigs in the All-Star game Friday night when the Minnesota Timberwolves visit Oracle Arena to face the Golden State Warriors.""continue": "<P>Love and Curry were<i>two of<b>four</b>first-time All-Star</i>starters when the league made the announcement on Thursday.</P><P>Love got a late push to overtake Houston Rockets center Dwight Howard in the final week of voting.</P><P>\"I think it's a little sweeter this way because I really didn't expect it,\" Love said on a conference call. \"I was already humbled by the response the fans gave me to being very close to the top (frontcourt players). The outreach by the Minnesota fans and beyond was truly amazing.\"</P>"}

When you pass continue as a param into text for escaped-continue you are removing the <p> tags at that step. You can either use exslt node-sets with XSLT 1.0 and handle the nodes inside the replace-string template, or rewrite your escaped-continue to parse nodes and text and only call replace-string for the text nodes.