This is a pretty simple one, but it wasn't obvious to me. I had to ask about it on the Unity forums.
I was working on my animation in a 2D perspective by clicking on the view cube arrow to give me an orthographic camera. I'm not sure what I did, but my manipulator went missing. I couldn't move the objects around anymore. It turns out, I hit the "2D" button, which flattens it to a true orthographic camera, and then, for some reason, gets rid of your manipulators. Just toggle it off, and you'll get everything back, just fine!
Check out these pictures for a visual explanation of my issue: