This expression means that this place will be ok or suitable (for whatever the people are planning to do). It is a sentence that is normally followed by an action (for example, 'it seems as good of a place as any [to eat our meal/to watch the race/to run]'). In this case, 'of' is used to indicate the quality ('good') of the noun ('place'). This expression also indicates that the person saying this knows there might be other opportunities ('as any') but chooses to use this place. There is also a similar expression 'it is as good a time as any'.

I personally would not have used the word "of" in that sentence. I would have simply said, "It seemed as good a place as any." I don't know if that helps you understand it any better. The sentence is saying that the place the speaker is talking about is no worse than any other place for his purposes. However, it also implies that the place isn't necessarily great, either. For example: Kenneth: It's starting to get dark. I think we should find a place to set up camp for the night before we lose the light. Amanda: How about that clearing just ahead? It's not far from our water source. Kenneth: Sure. It's just as good as any other place on this section of trail.

This is more of an older/traditional way to speak for most, however, the "of" is just expressing the greatness of the place. Simply stating that the place is good like the other places, even temporarily. Even feeling lucky to possess such a place for the time being.

I agree with Melissa. In the UK, you'd hear people say "as good a place as any". In the US, the "of" creeps in, but I think it's a mistake (along the lines of the way "I could care less" is common in the US, but not in the UK, and actually means the opposite of what the speaker intends, which is "I couldn't care less"). There are lots of errors/additions/omissions/transformations in English as spoken by natives that make it harder to understand than necessary and this is a good example, IMHO. :D