Kami: I don't think that excerpt really explained the thinking behind the phrase, it was more a rundown of the meditation as a whole. And attributions are nice.
Anyway, Descartes's phrase can be interpreted as having been arrived at several ways- particularly when taken out of context (I think I recall him going on to explain himself, but I'll have to find that later), but if memory serves me the upthrust is that if he didn't exist there'd be nobody to do the thinking- you can't have a nonexistent reference point. Pretty straighforward if you think about it.
Something he overlooked (which was where I arrived at when I tried his procedure) is that existence is, basically, defined as what we see. After all, what definition for reality is there other than our universe? There could be another "real" world behind this one, hidden by a mass of complex illusions, but since we can't see it or know of its existence and it has no direct bearing on our reality, it might as well not exist, and postulating its nature is a futile and mildly stupid task, sorta like wondering what unicorn farts would smell like. As far as it matters, what we see is more or less what we get, and any reality beyond that is, at least for now, inaccessible- hence, our universe exists, or at least is more demonstrably "real" than anything else by a good deal (and what better measure of existence than that?).
This doesn't mean it exists in just the way we think it exists, of course. There always are and always will be almost infinite unknown variables- but until they act on something within our perception, their existence is entirely theoretical, and shouldn't be assumed.