Mathematical logic can be motivated by other areas of math in at least two different ways:

[1] It allows you to formulate (and prove) results about the unsolvability of certain problems. These are obviously essential, since they tell you that you shouldn't spend too much time trying to solve those problems, which are often very natural problems.

For example, as a group theorist, you might often want to know whether two particular groups given by generators and relations are isomorphic. It would be nice to have some set of tools that allowed you to solve the problem mechanically, but no such tools exist (Novikov's Theorem).

Or, as a number theorist, you might wish for a set of tools allowing you to decide effectively whether a given polynomial equation has integer solutions. This is ruled out by the Davis-Putnam-Robinson-Matiyasevich Theorem.

[2] It can give you easier proofs of theorems in seemingly unrelated subfields. I hope somebody can provide/confirm examples here...for example, I think Gödel's Compactness Theorem gives some mileage in algebraic geometry (Nullstellensatz?), and nonstandard analysis can simplify a number of proofs (Tychonoff's Theorem?). (Although nonstandard analysis isn't exactly mathematical logic, the fact that proofs of standard results using nonstandard analysis can be trusted is a theorem of logic.)