I am by no means the expert on algebraic geometry. But maybe I can say a little bit. Kontsevich seems ever wrote a book"Beyond Number" There is one paragraph:
“Very often a mathematician considers his colleague from a different domain with disdain -- what kind of a perverse joy can this guy find in his unmotivated and plainly boring subject? I have tried to learn the hidden beauty in various things, but still for many areas the source of interest is for me a complete mystery.
My theory is that too often people project their human weakness/properties onto their mathematical activity.
There are obvious examples on the surface: for instance, the idea of a classification of some objects is an incarnation of collector instincts, the search for maximal values is another form of greed, computability/decidability comes from the desire of a total control.
Fascination with iterations is similar to the hypnotism of rhythmic music. Of course, the classification of some kinds of objects could be very useful in the analysis of more complicated structures, or it could just be memorized in simple cases.
The knowledge of the exact maximum or an upper bound of some quantity depending on parameters gives an idea about the range of its possible values. A theoretical computability can be in fact practical for computer experiments. Still, for me the motivation is mostly the desire to understand the hidden machinery in a striking concrete example, around which one can build formalisms.
..... In a deep sense we are all geometers."
I think what Kontsevich mean is that not only the result should be correct, but also the method to get the result should be elegant and natural. Just as he mentioned, the most interesting things to him is the hidden machinery behind the striking examples. For example, say Atiyah-Singer index theorem. Rosenberg ever mentioned in the class, this theorem should have the machinery living in the abelian categories or even exact categories instead of triangulated categories(where Grothendieck Riemman Roch is now living). I guess what they are thinking about is that one should use some universal constructions ,universal theory"(in some sense). They always make emphasis on one sentence "Mathematics should be simple" which might means that the proof of some big theorem should be simple. That is to say, one does not pay much "brain thinking" because "brain of human are weak"
However, I agree with Emerton that this is very personal feelings
