Why the exp function. Let me take a crack at the first part of this. I refuse to apologize if I only produce a B movie.
Lie groups are manifolds, Lie algebras are vector spaces.
I don't know about you, but I think life is easier in the land of vector spaces than in the land of manifolds. So, consider the following problem in mechanics or field theory. It is not hard to see that the infinitesimal symmetries form a Lie algebra. So, for me, this gives a decent motivation for why onewants to understand Lie algebras on their own terms. One can also make a quantum mechanical version of the above example.