Phrase Meaning – What Does ‘Someone Is Just F-ing with Somebody Else’ Mean?

I think OP's specific context is a (very slightly) "abnormal" usage, where most native speakers would probably say "Just because" (i.e. - for no particular reason that I can articulate).

In other contexts, just like that is normally used to mean as quickly as that, or with no further deliberation. In such contexts, that references a short (or even, non-existent) antecedent discussion/preparation.

The British stand-up comic Tommy Cooper used "just like that" as a catch-phrase - usually as facetious wordplay (not like this, [just] like that!), after a bungled "sleight-of-hand" magic trick.

To catch up with X means to be up to date with X, either by getting information or performing outstanding tasks.

Caught up by itself can mean "stuck due to some circumstance outside of my control" Interestingly catch up in present/future tense doesn't ever mean this, but (will) get caught up by can be used as an equivalent. Got caught up is valid as well.

With X can be used with this sense of caught up which works well if X is a group of people and you temporarily joined or "got caught in" their group for a while unintentionally.

So this second meaning is likely intended.