What is the point of studying the cardinality of infinite sets? Is it just for the sake of completeness in the understanding of cardinality of sets in general?

Or is there any particular uses of such knowledge apart from further abstractions of the concept? 

Thank you!
"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.
First,
we say that
two sets have the same cardinality
if there is a 1-1 mapping
between their elements.

Then,
we can show that
there is no 1-1 mapping between
a set and the set of
all its subsets
(called its power set).
This means that
there are an infinite number
of possible cardinalities,
each gotten by
successively
taking power sets.

In particular,
since the integers are countable
(by definition),
the set of all subsets of the integers,
which can be shown to be equivalent
to the reals,
is not countable.