Among Lie groups, why study the semisimple ones?

Simple groups and simple algebras are "the ones all that matter" for an intrinsic reason. They are the atoms of this universe, so to speak, and hence are interesting. In case of Lie groups, there are furthermore reasons from geometry, why we are in particular interested in semisimple ones. However, even there, the reason is similar. We want to understand the basic "irreducible" components. One example illustrating this is the classification of Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, it makes sense to further restrict oneself to classifying the irreducible, simply connected Riemannian symmetric spaces. We end up with simple Lie groups in Case $A$, and with compact Lie groups in case $B$. But even for compact Lie groups we are very close to semisimple Lie groups, and one could argue that also there "the ones all that matter" are semisimple ones.

From the standpoint of representation theory, the larger class of reductive groups are "the ones that matter" (this is an oversimplification, as finite-dimensional representations of non-reductive groups are crucial to understand, even if you only care about representations of reductive groups). But you can't understand reductive groups without understanding semisimple groups first.