How I always think of it is the following:
the infimum of a finite set of strictly positive numbers has to be strictly positive, but the infimum of an infinite set of strictly positive numbers can be zero.
This fact leads to examples where the finiteness of the subcover has practical importance.
Lebesgue numbers are used to prove that every continuous function from a compact space is uniformly continuous. (See for example here).
If the set is not compact, then we cannot necessarily define a Lebesgue number (see here).
With compactness, if we have a positive number associated to every open set in the open cover, then we can always take a finite subcover and thus find a "smallest" number which is still greater than zero.
If we didn't have compactness, then the "smallest" number might be zero, and then the proof would no longer work.