This is a soft-question in the sense that I have no confusion over the definitions of a bundle, fiber (see: Wikipedia), trivializing open sets and/or a locally trivial rank of $r$ (below are definitions for the latter two)
A surjective smooth map $\pi:E\to M$ of manifoldsis said to be locally trivial of rank $r$ if i.) each fiber $\pi^{-1}(p)$ has a structure of a vector space of dimension $r$, ii.) for each $p \in M$ there are open sets $U$ of $p$ and a fiber-preserving diffeomorphism $\phi:\pi^{-1}(U)\to U\times \mathbb{R}^r$ s.t. $\forall q \in U$ the restriction $\phi\bigg|_{\pi^{-1}(q)}:\pi^{-1}(q)\to \{q\}\times\mathbb{R}^r$ is a vector space isomorphism. Such an open set $U$ is called a trivializing open set for $E$, and $\phi$ the trivialization of $E$ over $U$
But what I am interested in hearing is that what was/is the motivation i.) for the fibers/bundles/trivializations, ii.) to name the constructs the way they were? As a novice to both topology and differential geometry, the fibers/bundles etc. seem to come out of nowhere, so thinking the bigger picture, and the why are we doing this is rather difficult.
