Earlier quoted context omitted.
Whenever I pick up a maths book, I read all these symbols that have no explanation whatsoever - it's obviously self-indulgent wanking. I admit I don't actually read the book, though, I just open it. But I can tell the symbols aren't english, or any other language I learned before my mind closed, so I can tell it's dross.
Well name a philosophy book that is worth reading. Maths books don't claim to be anything but self-indulgent.
You can construe the cogito as an argument:
P1 I think.
P2 If I think, then I exist
C1 I exist.
P1 and P2 are premises, C1 is derived.
The interesting point is that P1 is not arbitrary, in the sense that if one grasps it, then it is true.
That's a really interesting property for a claim to have - especially for a contingent claim to have (e.g. 2+2=4 is necessarily true, and so if one grasps it, trivially it is true, but that's not interesting in the same way).
Further, D thinks that if one grasps it, then one knows one grasps it, (and that it is true).
So for D we have a priori certain knowledge of a contingent claim.
If course, you can question 2, and some people have. Quine is one person to read on that.
Re 'what does it mean to exist' - what do you mean by that? Your criticism demonstrates some ignorance of Descartes, as Descartes makes substantive claims about the nature of his existence as revealed by the reasonining in Mediations. For example D would say he knows that he exists as a thinking subject - a substantive claim about his nature. He goes on to make lots of controversial claims about his nature (some of which get us to Cartesian Dualism).