Wait, you have a Creative Cloud subscription, but you've been using CS6 version of After Effects? MY GOODNESS, MAN! You have been missing out on a LOT!
The Creative Cloud app might need to be updated...what version of it are you running? It should give you options to install other versions of software.
Keep in mind, you can have versions installed alongside each other. So you can have CS6, CC, CC 2014, CC 2015, and CC 2015.3 all installed alongside each other, if you want.
So...Wowzers. Um...if you jump straight to 13.8.1 (the latest version of CC 2015.3), make sure you read up on the new architecture! RAM preview is gone, for example. (It's not really gone, but it works entirely differently under the hood.)
all about previews in After Effects CC 2015 (this new preview engine has been greatly improved in the updates since 13.5 was first released, by the way, so not all of this is still true, but the basic idea is the same.)
features not available in After Effects CC 2015 (13.7) (multiprocessing, for example, is gone - to be replaced in the future with something very different, but better)
And the new architecture produced some problems at first. Most are fixed now, but here are the known issues in CC 2015.
So, yeah, the UI and the renderer are separated for the first time in AE's 20+ year history. Which means a better, more modern AE in a future version. (And 13.8.1 is my favorite version of AE ever.) It's very exciting.
Now that you've gotten the technical weirdness of the new architecture out of the way, and, since you're coming from such an old version, it wouldn't hurt to see some of the other new features you've been missing (now that Adobe has gone to a subscription-based model, point releases now include new features too, so you'll need to read about them):
There was also a useful blog series recently on some of the added features since CS6. A lot of these aren't splashy new features that people talk about, but are VERY useful additions to the workflow in AE.
I've put them in the order of my favorites!
Something people aren't really talking about is the new panel stacking/soloing business in CC 2015 and later. When you go to use it, choose the "All panels" workspace at first. It's GREAT for smaller screens! That's what I built my MacBook's workspace off of. (My Windows workstation has four monitors, so I have an entirely different workspace for it!)
I know this is a lot, but I'm just really excited for you. There are so many improvements to your workflow in your future!
So... How do I update after effects?