Using this expression, people put themselves in the center of the universe. Opening them to the possibility that they are the only things that are real. That, in my opinion, is a very arrogant way of looking the entire universe. It took humanity centuries to figure out that the earth was not the center of the universe, and here comes this weird French guy trying to convince us all that he is!
What Descartes says is that just because we think we see stuff doesn't mean it's real. It's arrogant to say that just because we think something is true, it's unquestionable. That's raising yourself to omniscience. That's what Descartes is saying, and so he establishes doubt about everything he perceives. The one thing that he can prove without his senses is that for there to be doubt there has to be a 'doubter'--himself, in this case. All he knows about himself is that he is something that can doubt. (and think.)
He's not telling us he exists, he's telling himself that he exists, and at the same time that if any of us exist (which he tries to prove) then we can only be certain that we exist. This doesn't make us the centre of the universe; he's establishing himself lower than you establish yourself: he says he could be wrong about everything except his own existence while you say you're right about your existence, my existence, his existence, and the existence of nearly everything you see. (I assume. If not you're making an awful fuss over something you seemingly agree with.)
Thinking does not prove existence. I've seen many people who haven't thought a day in their lives, and yet unfortunately... they still exist.
By the same logic, if today is Wednesday, that doesn't prove that today isn't Friday. I've seen many days that aren't Wednesday and yet, unfortunately... they still aren't Friday. (Monday isn't Friday, therefore Wednesday is Friday.) WRONG!
(Anyway, some people may not think a lot, but they certainly think. Any action whatsoever takes thought.)
In my opinion, establishing the universal doubt may be the greatest thing Descartes ever did in the field of philosophy, and his extremely lame (I can not put into words how awful it is) attempt to eliminate it was the worst. :-)