It means that I am up to date. Say for example, you are watching a TV show and you have seen all the episodes and you are waiting for the next episode which will be available next week. Then you can say that "I am caught up, to the latest episode on the show". To catch up, means to be up top date or have the latest information or doing something that makes you "latest". You can even do this while running. I have to "catch up" with those guys who are running ahead of me, or I will lose the race. Hope this helps you to understand the phrase "To catch up". "Caught up" is the past tense. "I will catch up" is future.

In these contexts, "caught up" refers to having finished or completed something, and now there are no more tasks to complete (at least for the current time). For example, in your first sentence, "caught up" means that Josh has finished everything that he had to do, and now that he has finished everything, he feels bored.