When you study measure theory, the importance of distinguishing different kinds of infinite sets becomes apparent. You want to be able to measure "reasonably nice" subsets of the real line in such a way that $[0,1]$ has measure one. If you have an infinite collection $E, E', ...$ of disjoint subsets, you want the measure of their union to be the sum of their measures. The measure should be translation invariant.
Without the knowledge that there exist uncountable subsets of the real line, this problem would seem impossible: all points have the same measure by translation invariance, and must all have measure zero in order for infinite sets to have finite measure. But then the measure of $[0,1]$ should be zero, and not one, because its measure is the sum of the measures of the individual points.
So, cardinality can be strongly motivated by some concepts in measure theory. But abstract measure theory did not come about until after Georg Cantor. What concepts or problems in the late nineteenth century could have motivated Cantor to come up with his results about uncountable sets?
Cantor was studying trigonometric series, and came across an iterative procedure (that he called the "derived set") that produced what we would call today an ordinal-indexed sequence. He had also studied constructions of irrational numbers in a paper that was cited by Dedekind when Dedekind introduced Dedekind cuts, and was interested in algebraic and transcendental numbers from a number theoretic point of view (which is the topic his thesis was in and his work before set theory was considered part of). I doubt his exact thought process is known, but those topics reasonably fit together to draw one to the countable/uncountable divide.
Cantor's "derived set" is what we would call "the set of limit points" or "the boundary" today. Other set theoretic notions, such as a set being closed, perfect, or dense can be (and were, by Cantor) defined based on this notion.
See HERE under the subheading "Mathematical Work / Number theory, trigonometric series and ordinals"