When something "gets caught", it means that it's not able to move freely. For example, if you're wearing headphones while you walk through a doorway, the wire might "get caught" on the doorknob. If you try to keep walking, the doorknob will pull the wire. Wires, clothing, hooks, ropes, and handles can all "get caught" on something. Here are some sentences that use "get caught":

A: Pull it tighter.

B: I can't! It's caught on something.

Her shirt got caught on a nail that was sticking out and ripped a little bit.

"Getting caught" is very similar to "getting stuck". However, "getting stuck" is a situation that's a little harder to get out of. If your foot "gets caught" on a branch while walking in the woods, you can easily move your foot around and release it. But if your foot "gets stuck" between two branches, it might take more force to pull it out.