Why is it interesting?

Fiber bundles are the appropriate mathematical tool to describe, for example, the physics around a magnetic monopole or also instanton effects. (This is described very nicely in chapter 1 of Topology, Geometry and Gauge fields - Part 1 Foundations by G. Naber). Moreover, non-local aspects like Gribov ambiguities can be understood much more clearly with fiber bundles.

They are also useful to show the similarity between general relativity and Yang-Mills theory, which we use in the standard model. The reformulation of the notions that we usually use in quantum field theory, like the notion of a quantum field, in the fibre bundle formalism, allows us to understand them geometrically. This makes them, contrary to what one may think at the beginning, much less abstract and lets us view at them from a completely new perspective.

See also: https://physics.stackexchange.com/questions/77368/intuitively-why-are-bundles-so-important-in-physics

Although gauge theory is introduced in the above inductive manner for historical and pedagogical reasons it is clear that the essential ingredients -the gauge potential, the gauge field, and the covariant derivative - have an intrinsic mathematical structure which is independent of the context.This structure has been well studied by mathematicians, in the context of differential geometry. In this context transformations g(x) are identified as sections of principal bundles, with Minkowski space J (as base and the Lie groups G as fibres, the scalar and fermion fields are identified as sections of vector bundles with base Jl, the gauge potential as a connection form for G{x), and i^„(x) as the components of the curvature. A review of these aspects is given by Daniel and Viallet (1980). The fibre-bundle formulation is not necessary for dealing with those aspects of gauge theory which are local in M, but it becomes important for understanding problems, such as the axial anomaly (next section) and the gauge-fixing ambiguity (Gribov, 1977; Singer, 1978) which are of global origin. It also shows that gauge theory, and thus the theory of strong, weak and electromagnetic interactions, is basically a geometrical theory. This is not only aesthetically pleasing but brings the unification of weak,electromagnetic and strong interactions with gravitation a step closer. Group Structure of Gauge Theories by Lochlainn O’Raifeartaigh

Layman?

The best laymen introduction to fiber bundles is "Fiber Bundles and Quantum Theory" by HJ Bernstein

Student

• A nice introduction to fiber bundles with many pictures is Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results by Adam Marsh
• A good textbook on the topic is Topology, Geometry, and Gauge Fields: Foundations by Naber

Researcher

Gravitation, Gauge Theories and Differential Geometry by Tohru Eguchi et. al.

Examples

Example1

Example2:

History

• VECTOR BUNDLES AND CONNECTIONS IN PHYSICS AND MATHEMATICS: SOME HISTORICAL REMARKS by V. S. Varadarajan