I've been playing AP Kennen for awhile now with many people recommending I use Zhonya's Hourglass. I'm not sure how to use item effectively so I was hoping for some advice. The only thing I could guess would be to use it after I use my ultimate in a team fight. Would this be a good idea and/or is there any other ways to use it well?
You're right. The best way to use Zhonya's Hourglass is by using it in a teamfight. Lightning Rush in, try to hit as many targets as you can, then activate Electrical Surge, then Slicing Maelstrom. Here is the description for Kennen's ultimate, along with the description for Zhonya's hourglass, from the League of Legends wikia page.
Add this to your two marks of the storm on each opponent with your E-W combo, along with any Marks of the Storm added from your ultimate before using Zhonya's hourglass. You can potentially stun all opponents.
Since Kennen's ult is very powerful due to its utility, enemies will try to focus you down to avoid stuns. That's when you want to Zhonya's. When it's active, you will continue to stun enemies.
Because of Kennen's lackluster AP ratios, many summoners will disregard Kennen in teamfights until after the salvo of stuns has been unleashed because he will not be anywhere near as much of a threat as the AD, for example. I've found that waiting to use Zhonya's until I start to take some damage is a better option than clunking down with it immediately, that way you are able to get an extra stun or two off with your combo, as well as disrupt the enemy team even moreso by denying them the chance to focus on your team (because you're in their face stunning all of them), and denying them the chance to focus on you (because you're invincible for two seconds).
Rule of thumb: Only use Zhonya's once you start taking serious damage.