What is the point of studying the cardinality of infinite sets?

"Studying the cardinality of sets" means "studying when there is or is not a bijection, injection, or surjection from one set to another" and it is one useful way of telling sets apart.

One early illustration of this is due to Georg Cantor, who used cardinality to give a simpler proof of something already known, the existence of transcendental numbers. Namely, he showed that there is a surjection from the set $\mathbb N$ of natural numbers to the set $A$ of algebraic numbers, but there is no surjection from $\mathbb N$ to the set $\mathbb R$ of all real numbers; whence it follows that $A\ne\mathbb R,$ i.e., there are real numbers which are not algebraic.

If I told you that every set of reals is Borel, how would you prove or disprove this?

Once you study cardinalities, you learn the Borel sets have $\aleph_1$ "levels", each level with $2^{\aleph_0}$ sets; therefore there exactly $2^{\aleph_0}$ sets of reals which are Borel sets. Being an educated man, you know of Cantor's theorem and you know there are $2^{2^{\aleph_0}}$ sets of reals. So definitely not all the sets of reals are Borel sets.

Well, what about Lebesgue measurable sets? Maybe not all sets of reals are Borel; but are all the sets which are Lebesgue measurable Borel?

Again, the answer is no, because of a simple counting argument: the Cantor set is Borel, and null. So every subset of the Cantor set is Lebesgue measurable. Again, there are only $2^{\aleph_0}$ Borel sets, but there are $2^{2^{\aleph_0}}$ subsets of the Cantor set.

You can do these games again with functions which are continuous (or rather close to being continuous, i.e., Borel measurable). It tells you that in the grand scheme of mathematical objects, the ones we care about are usually "the pathological exception" and not the other way around.