Physicists love to talk about that stuff, but no one ever has the guts to explain it into enough detail to show its beauty!!!

Perhaps the wisest thing is to just focus entirely on the $U(1)$ part to start with, which is the quantum electrodynamics one, which is the simplest and most useful and historically first one to come around.

Perhaps the best explanation is that if you assume those internal symmetries, then you can systematically make "obvious" educated guesses at the interacting part of the Standard Model Lagrangian, which is the fundamental part of the Standard Model. See e.g.:

- derivation of the quantum electrodynamics Lagrangian
- Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 7 "Interaction Theory" derives all three of quantum electrodynamics, weak interaction and quantum chromodynamics Lagrangian from each of the symmetries!

One bit underlying reason is: Noether's theorem.

Notably, axelmaas.blogspot.com/2010/08/global-and-local-symmetries.html gives a good overview:
so it seems that that's why they are so key: local symmetries map to the forces themselves!!!

A local symmetry transformation is much more complicated to visualize. Take a rectangular grid of the billiard balls from the last post, say ten times ten. Each ball is spherical symmetric, and thus invariant under a rotation. The system now has a global and a local symmetry. A global symmetry transformation would rotate each ball by the same amount in the same direction, leaving the system unchanged. A local symmetry transformation would rotate each ball about a different amount and around a different axis, still leaving the system to the eye unchanged. The system has also an additional global symmetry. Moving the whole grid to the left or to the right leaves the grid unchanged. However, no such local symmetry exists: Moving only one ball will destroy the grid's structure.Such global and local symmetries play an important role in physics. The global symmetries are found to be associated with properties of particles, e. g., whether they are matter or antimatter, whether they carry electric charge, and so on. Local symmetries are found to be associated with forces. In fact, all the fundamental forces of nature are associated with very special local symmetries. For example, the weak force is actually associated in a very intricate way with local rotations of a four-dimensional sphere. The reason is that, invisible to the eye, everything charged under the weak force can be characterized by a arrow pointing from the center to the surface of such a four-dimensional sphere. This arrow can be rotated in a certain way and at every individual point, without changing anything which can be measured. It is thus a local symmetry. This will become more clearer over time, as at the moment of first encounter this appears to be very strange indeed.

axelmaas.blogspot.com/2010/09/symmetries-of-standard-model.html then goes over all symmetries of the Standard Model uber quickly, including the global ones.

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