To "get tied up with" something means to become really busy with something that you aren't able to stop doing. People usually "get tied up" at work. They get a lot of work that needs to be done quickly. People can also "get tied up" with their kids, doing home repairs, or other things:

Oh, is it eleven o'clock already? I got so tied up with sending out these invitations that I didn't even notice.

If you tell people that you're "tied up", they'll understand that you're too busy to do anything else. For example, if you call someone and they say:

Sorry, I'm kind of tied up at the moment. Can I call you back?

...you should understand that they're too busy to talk now.