It's very simple: you'd use <xsl:processing-instruction>
if you needed to output a processing instruction in your output XML. If you have no need for PI's in your output, then you don't need the processing-instruction element.
As to why you might need a PI in your output, that depends entirely on what your output will be used for.
I've used them in the past to add <?xml-stylesheet>
instructions into my output:
<!-- Link to the stylesheet for people who wander in. -->
<xsl:processing-instruction name='xml-stylesheet'>
type="text/xsl"
href="<xsl:value-of select='$stylesheet'/>"
media="screen"
</xsl:processing-instruction>
produces:
<?xml-stylesheet type="text/xsl" href="http://nedbatchelder.com/rss.xslt" media="screen"?>