Like many properties, this is a notion of smallness. It's not about the smallness of a subset $U$ of $X$, but smallness of a collection ${\mathcal C}$ of subsets $U$ of $X$.

You may consider a well choosen sample of neighborhoods (the sets $V\in{\mathcal T}$) and ${\mathcal C}$ ought to be finite with respect to that sample (finite pro $V$).

• A topologal space is paracompact if it any cover has a refinement with that property.
• The sample of $V$'s above may be very big, so ${\mathcal C}$ is really only small w.r.t. the sample. In a compact space, on the other hand, the cover itself is finite (and you don't need to consider that sample).
• Note that the name locally compact is already used for the situation where every point $x\in X$ has a compact neighborhood $V$.

Wikipedia: Locally finite collection, Paracompact space

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Topological space

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