Anselm’s Argument presupposes that God is, as (almost) all men have always thought him to be, that than which no greater can be conceived – the Ultimate Being, along all dimensions of perfection, virtue, & goodness – and goes on to argue that such a being that actually exists, being greater than the mere idea of such a being, cannot but be what we mean by “God.” So, if we are talking about God, we *must* be talking about a being who actually exists. If we are talking about a being who might not or does not actually exist, we are not talking about God in the first place, and so all our talk of that thing (whatever it is) is simply inapt, ergo moot. Since we are in fact talking about God, we must be talking about a being who actually exists. So, in order for us even to talk about him, God must actually exist. The actual existence of God is implicit in the concept of God. This is what Aquinas is getting at in arguing that the existence of God is essential to his nature.
Almost all of Anselm’s critics lose him at that point. His famous Argument – which is perhaps the most famous in the entire history of philosophy – seems to many, if not most, to consist entirely of vain logic chopping; of proving a conclusion by a prior definition, that might with equal reason have been defined some other way.