I'll be brief with the summary of the argument, but I want to focus on premise (1), which is really the only controversial premise of the argument.
1. Possibly, a maximally great being exists. (Premise)
2. Necessarily, a maximally great being is maximally excellent in all possible worlds. (Premise)
3. Necessarily, a being is maximally excellent if and only if it is omnipotent, omniscient, and morally perfect. (Premise)
4. Therefore, a maximally great being exists. (From 1 - 3 and S5)
Here's an argument for premise (1):
5. Necessarily, an imperfection can only be known if what is perfect (maximally great) is intelligible. (Premise)
6. Imperfections are known. (Premise)
7. Necessarily, whatever is intelligible is possible. (Premise)
8. Therefore, a maximally great being possibly exists. (From 5 - 7)
Of course, this is just a summary of the argument. Please take that into account.