Like mentioned by @JofryHS, css doesn't really respond to clicks. 
Using inline js click handlers isn't great as it'll leave you having to repeat yourself a lot, but from the sounds of things you're fighting against not having enough access to Sharepoint (sigh, corporate networks).
This uses an inline click handler to show the answer: 
<div class="question" id="q1">
    What colour is the sky?
    <button class="answerButton" onClick="document.getElementById('q1Answer').style.display='block'">Answer</button>
    <div class="answer" id="q1Answer">
        Overcast and grey, because I live in the UK.
    </div>
</div>
<div class="question" id="q2">
    Why does it always rain on me?
    <button class="answerButton" onClick="document.getElementById('q2Answer').style.display='block'">Answer</button>
    <div class="answer" id="q2Answer">
    I'd have thought you figured this out already... It's Britain, of course it always rains on you!
    </div>
</div>
the answers are hidden using css (note the display:none inside the .answer block): 
.answer {
    display:none;
    background-color:#e5e5ff;
    padding:15px;
}
.question {
    background-color:#f5f5f5;
    padding:15px;
    border-radius:5px;
    margin:7px;
}
Essentially, all the onClick does is change the css of the answer from display:none to display:block, rendering the answer visible. 
There's also a tad of css to make it look shinier, and here's a demo jsfiddle