I am reading Kleene's "Introduction to Metamathematics" and I got to the chapter "Recursive functions". I am learning logic in order to understand set theory in order to understand topology. Do I gain something from recursive functions? Is there application of recursive functions in other mathematics areas or is it purely necessary for proofs about formal systems. What is the use of recursive functions in logic? Why is it interesting to learn more about them?
Headline response. If what you want to get to know about is set theory, then you don't need to know about recursive function theory.
Slightly longer response. Yes, if you want to understand the general idea of a formalized theory (like formalized set theory), you need to grasp e.g. the (informal!) idea of its being effectively decidable what's an axiom, what counts as a proof, etc. But to grasp that, all you need is something like the idea of its being decidable-by-a-suitably-programmed-computer what's an axiom, etc. (or something like that). For these purposes, you don't need a theory of computable functions or the idea that the computable functions are just the recursive ones. So you don't need to know about recursive function theory.
One more (important!) remark. Kleene's book -- over 65 years old -- is a marvel. It is still worth reading if you already know an amount of logic. But it certainly isn't the place for you to start now, with decades of later logic books orientated to beginners to choose from. But how to choose? Here is an annotated Teach Yourself reading guide through some of the logical literature linked here, including suggestions of introductory books on set theory (and if you take a look at them, you'll see how little logic you actually need to get started on set theory).
This is suppose to be a comment. But what I understand these days is that: first order logic is kind of hard to define "infinity", or what you may write in the proof as "...". Notice "..." is sort of illegal in set theory, so recursive is a methods to deal with it.
Also, It was used to define the theroems, which comes handy when dealing with define operations like multiplication and addition.