@KripkePlatek This is a bit too vague of a question for me, I think. As 21820 explained, the transitive closure of a set x is defined to be the smallest transitive superset of x. Assuming x has a transitive superset at all, then you can define the transitive closure as the intersection of all transitive supersets. You can also define it recursively with unions, like how you wrote it, and you can prove that those two definitions are the same.

As for its "utility", in the context of what you wrote, I'd say it's used to make transitive models. If you want to build a model for set theory, and you want it to include some set x, then you probably also want to include the transitive union of x. At the very least, via Extensionality, the identity of x is informed by its members, so if we include x we should include its members. Similarly we should include the members' members, and the members' members' members, and so on.

So we end up including the whole transitive union. If we don't do that, then there would be some set x such that we do not include all its members. Then we can form a strict subset y of x such that y contains exactly the members of x which are included in the model. Now x and y are indistinguishable from the perspective of the model, in the sense that they have the same members (in the model), but they are not actually equal, so Extensionality is violated (undesirable).

correction: there would be some set x in the model such that we do not include all its members.

Additionally, by Mostowski collapse, any wellfounded model of sets is isomorphic to a transitive model, so there is no reason not to use transitive models.