I do have an intuitive understanding of compactness based on Euclidean space but not so much for paracompactness. Based on the Heine-Borel theorem, for a subset $S$ of Euclidean space $\mathbb{R}^n$, the following two statements are equivalent:
Which gives an intuitive idea of compatness. I would highly appreciate if someone gave a pictorial and an intuitive idea of paracompactness.
There is no easy intuition for paracompactness, I'm afraid. All metric spaces are paracompact, and these can be very different. All compact spaces likewise (a finite subcover is trivially a locally finite refinement).
The property was first introduced by Tukey under the name "fully normal" and only later it was found that there other so-called covering properties (properties of the form "every (bla) cover of $X$ has a (bla2) refinement (or star-refinement or subcover), for different "values" of bla/bla2)) were mutually equivalent or had interesting behaviour w.r.t. products/subspaces etc. and paracompactness emerged as the most interesting or "natural" variant of such properties. Dieudonné was the first to use the locally finite refinement formulation of the definition. Then it was found that there were nice theorems that were connected to uniform spaces and the general problem of metrisability of a space in which notions closely related to paracompactness were used too. It really took off as a popular property when it was found that in order for all open covers to have a partition of unity subordinated to it, we need the space to be paracompact (under mild suppositions) and these were used to "glue together" local functions on manifolds (where we can use coordinate chart covers) to global ones (for integration e.g.) So thus it became a popular condition in manifold theory. For Euclidean manifolds paracompactness is equivalent to being metrisable (globally).
Hope this helps somewhat as a motivation.
