As a physics student I can easily understand the motivation for studying manifolds and why the definition looks the way it does, I only have to think of Minkowski space in GR. But for the life of me the notion of a fibre bundle seems highly unmotivated. Why would we want to talk about manifolds in this strange way? Why are we jumping through hoops and ladders to talk about a manifold locally as a product space of two things?
When constructing or defining a mathematical object, you have to strike a certain balance: You want your object to be general enough to have lots of applications but structured enough to let you answer important questions. Fiber bundles are very general (in the sense that many spaces can be viewed as fiber bundles) and very structured (in the sense that you glean a lot of information about maps into and out of a space which decomposes as a fiber bundle).
As a physics student who has studied some GR, you can understand the importance of local coordinates. Fiber bundles provide another sort of local picture, decomposing your space into subsets possessing a certain symmetry. More importantly, perhaps, they give you some idea about how these local pieces glue up to a global structure. For physicists, vector bundles (like the tangent bundle) are an especially rich source of useful fiber bundles. Physicists care a great deal about "connections" on bundles, which are ways to compare points in nearby fibers. For example, connections give a way to understand parallel transport in GR. Also, much of classical mechanics can be formulated using the language of vector bundles.
When I switched from physics to math, it took me a while to appreciate that manifolds aren't simply spaces in which you should imagine some creature living. They often represent some additional structure on another space, such as the space of "directions" on a manifold (i.e. tangent bundle) or a space whose elements are loops in a manifold.
Because then we don't have to worry ourselves with the nature of the ambient space; that is, the manifold needn't be embedded in a higher-dimensional space for us to talk about it.