I know that we are able to classify all the finite simple group, but I don't understand how to use this classification to prove results like that every simple group has two generators.
The difficulty is that we still have an infinite number of such groups ($ A_n $ for $ n \geq 5 $), so it seems to be very problematic to varify a conjecture via loop on all of them.
I guess I miss some important assepcts on the classification... So how it works?
I'm not an expert, but at a very naive level the theorem can be applied in the same general way as any other classification theorem: by reducing the problem to a set of cases which are easier to analyze.
The point is that knowing that a group lies in a certain family gives us additional techniques for studying it - e.g. there are properties the alternating simple groups have, which finite simple groups don't in general. So if you want to prove something for all finite simple groups, you can break it down into a series of subproofs: one proof for each of the finitely many infinite families in the classification, together with a family of verifications for the sporadic groups individually.
That said, to the best of my knowledge the CFSG doesn't make anything easy - establishing that a result holds in any given case (even the individual sporadic cases!) is usually a huge task in and of itself (as you say, it can't be done by checking each individual group in the family), and indeed may need a further classification theorem breaking that individual family into finitely many sub-families. But the CFSG lets us "tie off" things once these results have been proved.
For what it's worth, you don't necessarily need to treat each infinite family separately - or, to put it another way, you only need a weaker classification theorem such as "all but finitely many of the (nonabelian) finite simple groups are either alternating or of Lie type," where we only want an asymptotic result so don't care about exactly what the sporadic cases are.