St. Anselm actually gave two ontological arguments he presented in the Proslogium, specifically in chapters 2 and 3. I'll explain the second version, which I think is easier to understand and more persuasive.
For, it is possible to conceive of a being which cannot be conceived not to exist; and this is greater than one which can be conceived not to exist. Hence, if that, than which nothing greater can be conceived, can be conceived not to exist, it is not that, than which nothing greater can be conceived. But this is an irreconcilable contradiction. There is, then, so truly a being than which nothing greater can be conceived to exist, that it cannot even be conceived not to exist; and this being you are, O Lord, our God.
I think this argument can be summarized like this:
Definition 1: God is a being that which nothing greater can be conceived.
Axiom 1: Necessary being, i.e. something that cannot be conceived as failing to exist, is greater than contingent being, i.e. something that can be conceived as failing to exist.
Premise 1: If God can be conceived as not existing, then we can conceive of something greater than God, following Axiom 1.
Premise 2: We cannot conceive of something greater than God, by definition 1.
Conclusion: God cannot be conceived as not existing, by modus tollens.
Now, obviously, I am not a philosopher and know nothing about the history of grappling with this argument, but:
Redditor’s summary seems to differ from the translation of Anselm insofar as its conclusion is about human psychology, whereas the ontological argument seems to be trying to say something about objective reality. It’s perfectly fine, and probably not very interesting, to argue that God cannot be conceived as nonexisting (although the immediate question — not sure whether philosophical or psychological — is whether anything can be conceived as nonexisting; whether the mere act of thinking about something by necessity puts it in some hypothetical plane where it is regarded as if it actually existed) — but the argument does not offer a link between our cognitive ability and something’s actual existence anywhere apart from our mind. This, I think, is what Dawkins and others who scoff at the ontological argument mean to say.