I'm not sure about the mathematical origins, but the original physical motivation was Yang and Lee's attempt to deal with the approximate SU(2)-symmetry of nucleons (protons and neutrons). The big step was (as I understand it) when Gell-Mann (and Ne'eman, independently at about the same time) realized that a diagram labeling experimentally observed particles was the weight diagram for SU(3). He made some [predictions][1] at a conference:

> following the presentation on Strong
> interactions of strange particles by
> G. A. Snow, both Ne'eman and Gell-Mann
> raised their hands to ask for
> permission to speak. The chairman
> called Gell-Mann, who was the more
> eminent physicist of both, and
> Gell-Mann announced that "[...] we
> should look for the last particle
> called, say, Ω-, with S=-3, I=0.
> [Here, I is isospin.] At 1685 MeV it
> would be metastable and should decay
> by weak interaction [...]"

and the rest was the [eightfold way][2].

Of course, principal $G$-bundles and the connections on them had been around for quite some time before (Simons famously pointed this fact out to Yang).


  [1]: http://backreaction.blogspot.com/2006_09_01_archive.html
  [2]: http://en.wikipedia.org/wiki/Eightfold_Way_%28physics%29