Possible: In this argument, "possible" means "logically possible." As long as there are no contradictions in a thing or idea, then it is logically possible, even if not actually true. It is logically possible that people can fly like Superman. There are at least no prima facie contradictions in that idea (since there are several movies clearly depicting it), and if someone claimed that there were, the burden would be on them to provide an argument. Therefore, "flying like Superman" is false, but still possible.
Necessity: Logically necessary. "No bachelors are married" is a necessary truth. It can't logically be false. Many philosophers of mathematics think that numbers actually exist. If so, they too are necessary. It is impossible for the number 3 to not exist.
Impossible: Logically impossible. The opposite of logically necessary. It is logically impossible that there are triangles with four sides. It is not logically impossible that a human can live without a head, even though such a thing is physically impossible.
II. Possible Worlds
Think of these as simulations of alternate realities. Ways that reality could be, one of which matches the actual world. These are conceptual tools used to describe what is (logically) possible, impossible, and necessary. So when we say something is logically possible, what we mean is that such a thing exists in some possible world, even if it doesn't exist in the actual world. And if something is logically necessary, then it exists in all possible worlds, including the actual world.
- For example, since there doesn't seem to be any logical incoherence in the idea of Superman, then we would say that Superman is logically possible and so he exists in some possible world. Clearly, he doesn't actually exist in the real world, however.
- That all bachelors are unmarried is a necessary truth, and so is true in all possible worlds. If numbers actually exist, then the number 3 is a necessary being and exists in all possible worlds. There is no possible world that has the numbers 1, 2, 4, 5 and so on, without the number 3.
III. A Maximally Great Being
The word "God" is often used as a code word for the creator of everything other than himself, and thus who is greater than anything else. There is nothing else greater than the Maximally Great Being, by definition. One property the MGB would have is necessary existence, and not possible existence. If it only had possible existence, then it would exist in some possible worlds and not others. But if could exist or not exist, then it would depend on something else for its existence (parents, for example), and thus would not be what it is by definition: the Maximally Great being. Or another way to think of it is that if it is the Creator of everything other than itself, and since every possible world contains something, then it would have to exist in every possible world.
IV. Premise 1: It is logically possible that the MGB exists
That is to say, it is not logically incoherent. This is for the sake of argument. Perhaps such a thing is logically incoherent, but go with it for now that there are no logical contradictions in the idea. That doesn't mean it actually exists, just that it is logically possible. You can accept that Superman is logically coherent without actually accepting that he exists. Do the same here for now.
V. Premise 2: If it is logically possible that the MGB exists, then MGB exists in some possible world
Remember what "possible" means, in section II above. If the MGB is logically coherent, then it is possible and thus exists in at least one of the simulations of alternate realities (possible worlds).
VI. Premise 3: If the MGB exists in some possible world, then the MGB exists in all possible worlds
Since as noted in section III, the MGB is a necessary being. If it were only a possible being, then it would not be maximally great. And as noted in section II, saying that something is necessary means that it exists in all possible worlds.
V. If the MGB exists in all possible worlds, then the MGB exists in the real world
Since the real world is one of the possible worlds. The conclusion is therefore that the MGB, or what some might label "God", actually exists.
VI. Does this actually prove God's existence?
Nope! What it does seem to do, however, is prove that if the MGB is not logically incoherent, then it follows that it must exist. In other words, the probability of the existence of the MGB is either 0%, or 100%. To deny the conclusion, you need to maintain that the MGB is logically impossible; that it contains a logical contradiction, and hence does not exist in any possible world. You need to maintain that premise 1 (section IV) is false. "Weak atheism" is out, otherwise you fall prey to premise 1 and the conclusion of the argument.
See here for an argument for the possibility of the MGB.