It may seem obvious that -1 × -1 = 1, but is it really as intuitive as it seems? In fact, really, it isn't nearly as obvious as it may appear and the proof can be nowhere near as rigorous as you would normally require one to be but it has to suffice because of how 'basic' the idea is.
The proof relies on the distributive law of arithmetic, which is that a(b + c) = ab + ac, this is so obvious to most of you that it may seem redundant even to state it but it is from this simple fact that the proof lies.
We will show that -1 × -1 = 1 by contradiction. We can obviously state that -1 × -1 = 1 or -1, then we assume that -1 × -1 = -1 is true. Consider -1(1 - 1) then by the distributive law we have that this equals to: -1 × 1 + -1 × -1 which from our assumption is -1 - 1 = -2. But this would then imply that -1 × 0 = -2 which is contradictory therefore -1 × -1 = 1 (which would make the equation we considered hold true, try it yourself).
This is a rather difficult proof to fully trust because we have made a pretty huge assumption, why must -1 × -1 = 1 or -1? Well, essentially, it doesn't have to but it just seems logical that it would be. As it is such a fundamental part of mathematics it is essentially defined by us in order for fundamental laws to continue to work. So if you find that the 'proof' I have given you is not enough then that's fine, it is true simply because it functions correctly..