You cannot update elements which are not rendered , rendered=false "is a JSF way to" to remove elements from the DOM Tree ,
its not like css display:none or visibility:hidden <- this two will keep the elements in the DOM tree but hidden , while the JSF rendered=false wont even render (keep) the element in the DOM tree (you wont even see it in the "view source" of the page)
So in you case you need to wrap the panelGroup with another `panelGroup' and update the id of the wrapper
<h:commandButton id="showButton" value="#{bean.wasPressed ? 'Hide' : 'Show'}">
<f:ajax listener="#{bean.toggle()}" render="explanationWrapper showButton" />
</h:commandButton>
<h:panelGroup id="explanationWrapper">
<h:panelGroup id="explanation" rendered="#{bean.wasPressed}">
<h:outputText value="something" />
</h:panelGroup>
</h:panelGroup>
also look at similar question
Can you update an h:outputLabel from a p:ajax listener?