I've been reading through Enderton's logic, this notion is introduced and is given special attention as it's said that they are crucial in the proof of incompletness theorems.
I grasp the formal definition, but what idea does this notion try to capture and formalize?
I think that a relation is representable in a given theory means that we can express the condition of membership of this relation using a sentence in that theory. Is that intuition valid?
Yes, that's the intuition you want -- except for minor terminology quibbles. You represent a relation with a formula with free variables for each position in the relation, which is specifically not a "sentence".
Note that there's a difference between representing a relation in a language with a particular interpretation (for example the language of arithmetic interpreted in the integers), and representing a relation in a particular theory.
In the former case, it is enough that you have a formula that happens to have the right truth value when you evaluate it in the integers.
The latter case is more subtle. Here the relation we want to represent is a given relation between the actual integers, but representability requires the the theory can prove that the relation is true or (as appropriate) false whenever we insert numerals into our formula -- independently of any particular model or interpretation of it.
So, for example, a relation that is representable in PA will always be representable in the structure $(\mathbb N,0,1,{+},{\times})$, but the converse is not always the case.