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I thought that Anselm was the guy who popularized this argument. Am I wrong?

No, not necessarily; though I think it's safe to say that the two men applied the insight differently.

In Proslogion, Anselm ((1033–1109), Archbishop of Cantebury, said something that Kurt Gödel likely agreed with:

"O Lord, you are not only that than which a greater cannot be conceived, but you are also greater than what can be conceived."

But I don't believe he was trying to advance a mathematical proof of the existence of God. He was simply saying that faith in God is reasonable.

Anselm opens his Monologion with these words:

"If anyone does not know, either because he has not heard or because he does not believe, that there is one nature, supreme among all existing things, who alone is self-sufficient in his eternal happiness, who through his omnipotent goodness grants and brings it about that all other things exist or have any sort of well-being, and a great many other things that we must believe about God or his creation, I think he could at least convince himself of most of these things by reason alone, if he is even moderately intelligent."

Anselm hopes to convince “the fool,” that is, the person who “has said in his heart, ‘There is no God’ ” (Psalm 14:1; 53:1) that there is, indeed, a God; and that this fact can be established by reason.

It has been pointed out that there are at least two ways to misunderstand what Anselm meant by his motto, fides quaerens intellectum — "faith seeking understanding."

First, Anselm isn't at all interested in replacing faith with understanding: If one takes ‘faith’ to mean roughly ‘belief on the basis of testimony’ and ‘understanding’ to mean ‘belief on the basis of philosophical insight’ [or arguably, on the preferred method of science, on the basis of direct observation and "falsification"], one is likely to regard faith as an epistemically substandard position; any self-respecting philosopher [or scientist] would surely want to leave faith behind as quickly as possible.

But as already mentioned, Anselm is not hoping to replace faith with understanding. Faith for Anselm is more a volitional state than an epistemic state: it is love for God and a drive to act as God wills. In fact, Anselm describes the sort of faith that “merely believes what it ought to believe” as “dead” (Monologion 78).... So “faith seeking understanding” means something like “an active love of God seeking a deeper knowledge of God.”

Or in other words, faith seeking understanding is seeking active relationship with and to God. By knowing Him, our understanding is increased — our understanding not only of Him, but also of the nature of His works.

The second common misunderstanding of fides quaerens intellectum is that, because it begins with “faith,” not with doubt or suspension of belief, it must be an inferior method of acquiring true knowledge of the world. This is the sort of approach roundly rejected by the scientific method — at least supposedly.

And yet it seems to me everything that science does is based on faith of some kind — just not faith in God.

To put it another way, the minimal faith of the scientist is that the world is intelligible, and therefore, can be understood by an intelligent being. Of course, the scientist does not/cannot ask what is the cause of, or the reason for, the intelligibility of the world. Such a question is never asked: It is utterly beyond the scope of the physical sciences. But it can be mathematically probed.

Kurt Gödel seems to have picked up on the Anselmian insight regarding God as "that than which a greater cannot be conceived, but ... also greater than what can be conceived," and recognized its axiomatic character. An axiom is an irreducible statement or proposition which is regarded as being established, accepted, or self-evidently true; in mathematics, it is a statement or proposition on which an abstractly defined structure is based. Thus an axiom falls clearly within the realm of mathematical structures that can be mathematically proved.

As Brian Wilson writes, "the mathematical model composed by Gödel proposed a proof of the idea [of God]. Its theorems and axioms — assumptions which cannot be proven — can be expressed as mathematical equations. And that means they can be proven."

And computer scientists Christoph Benzmuller and Bruno Wolzenlogel Paleo evidently believe that is precisely what Kurt Gödel managed to do.