So the upshot is that a 1-1+onto or bijective function has a unique inverse.
Thus this is a way of extending the ordinary notion of counting elements to infinite sets. When we count finite collections, we are putting them into 1-1 correspondence with a subset of the integers. To extend that notion to infinite sets, two sets have the same cardinality or "size" if there is a bijection between them.
The cardinality of the even integers is the same as the cardinality of the integers. Why? Here is a bijection f(N) = 2N.
Every non-empty open subinterval of the real line, no matter how small, has the same cardinality as the whole real line. Why? Here is a bijection: f(x) = arctan(αx); with "α" some suitable scaling factor that maps the arbitrary interval into (-π/2, π/2).
To prove the reals do not have the same cardinality as the integers, produce an enumeration of the reals, then show there is always a real number it doesn't contain. That's Cantor's Diagonalization Theorem.
Here's another way, more abstract but actually less difficult. Define the powerset of a set to be the set of all subsets of a set. So the powerset of {1, 2} is the set {{1}, {2}, {1, 2}, {}} [It's called the powerset because if a finite set has "S" elements, the set of all its subsets has 2S elements.]
Show that there is no bijection between any set and its powerset. Cantor did this already. It's the so-called "who shaves the barber" proof. Then show that there is a bijection between the reals and the powerset of the integers. Since there's a bijection between the reals and the powerset of the integers, there can't be one between the reals and the integers themselves.
In this extended sense [that there is no bijection] there are "more" reals than there are integers.