It's a highly technical and (I think -- though of course I'm no Goedel) bullshit-rich version of St Anselm's ancient "ontological argument" for the existence of God. ("Ontological argument" is a pretty stupid name. Blame Kant.)

Outline:

0. Use modal logic. (Goedel's "proof" requires that you be able to say things like "for some x, necessarily P(x)", which -- if you think about modal logic in terms of possible worlds, which many people do -- means that you need to be able to think of particular objects existing in multiple worlds. It is far from clear that this really makes sense.)

1. Assume that all properties of things can be classified into "positive" and "non-positive" properties, and that a few boring technical axioms about "positivity" hold. Intuitively, the "positive" properties are supposed to be the ones it's good for something to have. (I see no reason to believe that there's any notion of positivity that's close enough to the intuitive one but that has the technical properties Goedel wants.)

2. Say that something is "godlike" if it has all positive properties. (We're aiming to prove that something godlike exists. Note that even if everything else works, this will only be a proof of the existence of God if "positive" really does have something like its intuitive meaning. That's why the tension between that meaning and all the technical requirements for "positive" properties is important.)

3. Theorems: (a) for every positive property, it's possible that something exists that has that property, and (b) in particular it's possible that something godlike exists. (The fact that these really are theorems, at least if you use a suitably chosen modal logic, is one reason why I think it unlikely that any notion of "positivity" exists that both satisfies Goedel's conditions and matches up with intuition.)

4. Say that x "essentially has property P" if x has P, and any other property Q that x has is a necessary consequence of P. (Kinda weird, but never mind.)

5. Say that x "necessarily exists" if all its essential properties are necessarily instantiated; i.e., for every essential property P of x it's necessary that there's something with P. (It seems to me to be stretching it to call this "necessary existence", but never mind.)

6. Claim that "necessary existence" is a positive property. (Seems pretty arbitrary. Anyway, this is the key point at which Goedel's "proof" makes contact with Anselm's.)

7. Now it turns out that we can put the pieces together and deduce that there necessarily exists something godlike.

The "proof" depends on taking a very fuzzy intuitive notion, that of something being "positive" or "good", assuming that it can be treated with the utmost formality, assuming that a bunch of highly technical assumptions apply to it (e.g., "if a property is positive, then necessarily it is positive" -- which only even makes sense if you go beyond first-order modal logic, and I'm buggered if I can see why it should be true), and then seeing what follows.

This is essentially the same procedure that yields the following (absurd) proof that God doesn't exist: If God existed, he could make something too big for even God to move; but if God existed, nothing could be too big for him to move; contradiction. This is, I repeat, absurd, but its absurdity is of just the same sort as Goedel's argument depends on.