I think the rotations in the Lorentz group just reflect the isotropy of space, which comes down to the quite natural idea that if you observe a physical system from a different direction it doesn't change its dynamics. The full symmetry group of special relativity is the Poincare group, which includes the spatial translations, reflecting homogeneity of space.

The gauge groups are interesting in being extra symmetries beyond the spacetime ones, and yet they're closely related to spacetime symmetries, e.g. SU(2) being the double cover of the rotations SO(3). I also find it interesting that the groups that are physically basic, such as SU(2) being the one required to represent the phenomenon of spin, are also mathematically significant, in this case since SU(2) is the unique simply-connected group associated with the shared Lie algebra of SU(2) and SO(3). That shows some kind of deep connection between mathematics and physics. I'm just at the beginning stages of learning QFT and differential geometry so I don't have a feel for why that is or what it means at an intuitive level, and haven't seen any explanation for it. I think at the moment it's just a feature of our deepest experimentally-verified theory and so it would need a deeper theory to explain it.