I think the best motivation for the theory of Lie groups and Lie algebras is representation theory. Like you don't need anything besides linear algebra to know what SO_n is, but if you want to know about how it can act on a vector space then you need to think about the Lie algebra.
The other great thing about Lie groups is you can discover new and valuable groups just from pretty basic topology. Like the Spin group, which you know has to be out there as soon as you know the fundamental group of SO_n, but otherwise would be very hard to think of.
The fancypants but I think most intuitive way to think about Galois theory is also with topology. It's an algebraic version of a much more geometric, visible story, the correspondence between {subgroups of the fundamental group} and {normal covering spaces}.
The other great thing about Lie groups is you can discover new and valuable groups just from pretty basic topology. Like the Spin group, which you know has to be out there as soon as you know the fundamental group of SO_n, but otherwise would be very hard to think of.
The fancypants but I think most intuitive way to think about Galois theory is also with topology. It's an algebraic version of a much more geometric, visible story, the correspondence between {subgroups of the fundamental group} and {normal covering spaces}.