Whenever I bench press I notice that the weight is transferring back onto my wrists. I feel I could go to a higher weight but slight pain when doing this is holding me back. Is this because of bad form (I googled this but wasn't sure) or, alternatively, do I need to get more strength in my wrists before this goes away?
When you bench press, it is very important to try and keep your wrists from bending back from the weight in your hands. In other words, you want to have your elbow inline all the way up with your knuckles. If it's too hard to do that with the weight you're using, than that's a sign that you're using too much weight.
This page should help you understand the correct form and technique for bench pressing safely. There is also a video of Mehdi demonstrating the correct form. If you'll pay attention to his wrist you will notice that he keeps them almost perfectly inline with his elbows.