Mathematical logic, and the associated notion of the existential quantifier, were invented only after Kant's time. Kant used other, more traditional concepts.
The ontological proof (or at least the version that Kant criticized) is related to the idea that God exists by necessity, that existence is an essential property of God. When Kant asserted that "existence is not a real predicate", what he meant was that existence cannot be an essential property of anything (that it was an inherently accidental property), and therefore cannot be an essential property of God.
Kant meant that existence was similar to, say, location. Joe can be today at New York and tomorrow at Washington. Joe's location would change, but Joe himself would not change. If Joe ceases to exist, in a way he himself would change.. but arguably his concept would not. That's why Kant went on to argue in terms of concepts and judgements, rather than in terms of entities and properties. He argued that predicating existence of a thing does not "enlarge" or "increase" that thing's concept. Relatedly, he argued that the judgement "x exists" is always synthetic rather than analytic (that is tautological, true by virtue of mere meaning).
Kant's proof, that existence is inherently accidental, is roughly as follows: Suppose that the existence of some A enlarges A. In that case, A and (A + existence) were different concepts. And then the proposition "A exists" would be necessarily false. Because if A exists, then it is actually (A + existence) which exists, and (A + existence) is, we assumed, different from A.
A hundred real dollars contain no more than a hundred possible dollars. For, as the latter indicate the conception, and the former the object, on the supposition that the content of the former was greater than that of the latter, my conception would not be an expression of the whole object, and would consequently be an inadequate conception of it. But in reckoning my wealth there may be said to be more in a hundred real dollars than in a hundred possible dollars—that is, in the mere conception of them. For the real object—the dollars—is not analytically contained in my conception, but forms a synthetical addition to my conception (which is merely a determination of my mental state), although this objective reality—this existence—apart from my conceptions, does not in the least degree increase the aforesaid hundred dollars.
(Critique of Pure Reason "Of the Impossibility of an Ontological Proof of the Existence of God")