This is a casual way of describing watching a TV show.
When you say "watch (a show)", it sounds neutral. When you say "catch (a show)", it sounds as if the person wanted to watch it.
You can also talk about "catching a show" when someone doesn't see all of it. For example:
I caught the first few minutes of it, but then I had to leave.
In addition to TV shows, you can "catch" things like:
- other people's conversations
- live sporting events
- songs on the radio