As has been said, Lie groups are our best theory encoding continuous symmetry. Lie algebra theory, which is the infinitesimal counterpart, is a theory good enough that numerous problems can be solved by look-up, rather than arguing from first principles. You can look at the history, particularly with Cartan and Weyl; you can look at the examples coming from "commutation relations" people want to study; you can look at representation theory or root systems the theory of universal enveloping algebras; you can look at string theory or the Langlands philosophy. It has been found very natural to look at the Lie algebra as a linearised object behind the Lie group, and something easier to study.

As has been said, Lie groups are our best theory encoding continuous symmetry. Lie algebra theory, which is the infinitesimal counterpart, is a theory good enough that numerous problems can be solved by look-up, rather than arguing from first principles. You can look at the history, particularly with Cartan and Weyl; you can look at the examples coming from "commutation relations" people want to study; you can look at representation theory or root systems or the theory of universal enveloping algebras; you can look at string theory or the Langlands philosophy. It has been found very natural to look at the Lie algebra as a linearised object behind the Lie group, and something easier to study.