The passage you quoted asserts that a certain infinite family of set-theoretic statements are all provable from the axiom system under consideration, probably ZFC. So it's describing an infinite family of set-theoretic theorems. The family contains one statement for each choice of the formula $\gamma(x,y)$. Namely, once you choose a formula $\gamma(x,y)$, the associated theorem is the part of your quotation from "Assume $<$ is $\dots$" to the final "$\dots$ for all $t\in A$."
It's reasonable to ask why Enderton doesn't just state a single theorem, that begins with "For all formulas $\gamma(x,y)$, if $<$ is a well-ordering $\dots.$" There are two reasons for that, but they both boil down to: That single statement wouldn't be a statement of set theory so it couldn't be proved in ZFC (or whatever axiom system is under consideration).
The first reason is a temporary one: I don't think Enderton's book contains any formal, set-theoretic definition of "formula", so "For all formulas" isn't formalized in the language of set theory. I said that this reason is temporary, because the notion of "formula" could be formalized in set theory (like just about any other mathematical notion); Enderton just hasn't done so.
The second reason, however, is permanent. Even if you formalize, in set theory, the notion of "formula" and then try to express Enderton's schema as a single statement, with $\gamma$ now being a bound variable (since it begins "For all $\gamma$") you'll need to express what it means for such a $\gamma$ to be true of particular sets $x,y$. This cannot be formalized in set theory. That is, the general notion of truth for set-theoretic formulas is not itself expressible as a set-theoretic formula. This fact is Tarski's "undefinability of truth" theorem, and it applies not only to set theory but to any sufficiently strong (and consistent) mathematical theory.
Because of this second reason, it's hopeless to try to convert Enderton's theorem schema into a single theorem of set theory; we're stuck with a schema.