It is simple.
We have two approaches to "measuring" the size of two sets. The first one is based on counting and it is the usual one we are accustomed to with finite sets : if the number of elements of set A is lesser than the numbe rof elements of set B (and this amounts to saying that A is a proper subset of B) then we say that B is Greater than A.
When we move to infinite sets, the issue is that we have no easy way to extend the usual way of counting the elements of a sets.
Thus, Cantor's intuition is that we can generalize the usual way of counting the elements of a set - i.e. defining a bijection between the set and a natural number - and founding it on the concept of bijection between sets whatever :
two sets (finite or infinite) are equinumerous when there is a bijection between them.
This new definition does not add nothing in the case of finite sets : two finite sets are equinumerous exactly when they have the same number of elements (the number we get counting them).
Cantor's new way of counting the elements of a collection is based on the notion of one-to-one correspondence, which is prior to (and independent of) the concept of counting number.
The basic result is that the new definition is applicable also to infinite sets and give us a way to measure their size.
The "unpleasant" result is that two infinite sets can be equinumerous also if one is a proper subset of the other.
This means that - for infinite sets - the well-know principle expressed by Euclid with :
Common notion 5. The whole is greater than the part,
is not applicable.
The fact that the use of the said principle may lead to problems with infinite sets was already known since at least Middle Ages; see the so-called Galileo's paradox.
So yes, the issue is related to
our understanding of infinite sets.
With infinite sets modern mathematics has stretched our intuition that is clearly based on our experience with "finite" facts.