One of the earliest motivations for going beyond Lorentz/Poincare symmetry was the realization that even though Poincare symmetry is one of the few things we need to construct a QFT, this is so general that tons of theories could potentially be constructed, so that if we have a given a set of fields the allowable interactions between the fields could take tons of forms and still respect Lorentz/Poincare symmetry, however in nature we actually only find a few types of interaction. The idea was that this might be because nature has more symmetry than we are aware of, and so people began looking for more symmetries.
A natural thing to do would be to look at the theories we have and ask if something is missing. In looking at Poincare symmetry, which is roughly 'Lorentz symmetry + translation symmetry', we note that QFT's use representations of Poincare, and that spinors can be used to build up all these representations of Poincare, however while we do consider spinor representations of Lorentz we don't consider spinor representations of the translations, we just use translations directly (e.g. as differential operators, eq. 4.89), so it may be the case that by adding spinor representations of the translations it will lead to a higher symmetry, which turned out to be called supersymmetry. This approach turned out to naturally by-pass some no-go theorems for how one would normally think to add extra symmetry in the way all other theories of physics would incorporate extra symmetry, giving a sense of uniqueness to the approach and kind of implying there are almost no other ways to add extra symmetry, hence part of the importance of the idea of supersymmetry and why it's studied so much.