(1) Though some proofs of the equivalence of the axiom of choice and well ordering use transfinite recursion, there are proofs of that equivalence that don't use transfinite recursion. More generally, transfinite recursion depends on the axiom schema of replacement, i.e., ZF. But the equivalence of the axiom of choice and well ordering is provable in Z alone.
(2) Examples of transfinite recursion include such things as recursion on the ordinals, such as the alephs, the beths, the cumulative hierarchy, etc. Also there are applications of transfinite recursion that are not just on the ordinals, but the applications on the ordinals might be the easiest to visualize.
(3) As to the statement of the theorem itself, remember that it is a theorem SCHEMA. It is stated in the meta-language for the object language of set theory. It is a statement that specifies that a certain set of sentences in set theory are all theorems.
What textbooks are you using?