"Set theory" is a vague term, which may refer to "naive" set theory that involves basic properties of point sets, functions, relations, etc. These are at the same object level as analysis, although they are more generally applicable and are used also in geometry, arithmetic, algebra, etc. Tao specifically refers to "set-theoretic foundations" however. These are axiomatic theories that "interpret" analysis by reconstructing it from what is provided in the axioms themselves, and nothing else. The most popular of them, ZFC (for Zermelo-Fraenkel-Choice, the last one refers to the axiom of choice), formally builds everything from a single predicate (belongs to) and a few objects provided by its existence axioms (empty set, an infinite set, the powerset of a set, etc.). Some of the constructs are then identified with analytic objects.
Constructing a meta-theory usually means axiomatizing a field, and reasoning about what can and can not be derived in the resulting formal theory, e.g. whether it is consistent, complete, its axioms are independent of each other, etc. A standard way of resolving these issues is constructing various set-theoretic models, e.g. Gödel proved (relative) consistency of the axiom of choice with other axioms of ZFC by constructing ZF models where it holds, Hilbert similarly proved the indepenedence of the parallel postulate from other axioms of Euclidean geometry. There is a similar way of axiomatizing real analysis, and the study of such axiomatization(s) is accomplished by formally embedding it into axiomatic set theory, see Is the axiomatic approach to defining R rigorous? This is why Tao considers "set-theoretic foundations" to be "part of the metatheory of analysis".