Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map, that has to also be alternating and satisfy the Jacobi identity.
Then, because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group whenever possible, to try and get insight and proofs in that simpler framework. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.
To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:
- the dimension
- the Lie bracket