To get this intuition, think of the integers. There are obviously an infinite number of them. Now think of the rationals. There are now an infinite number of numbers between 1 and 2, so there are more rational numbers than there are integers; in fact infinitely more.
So, in set theory, cardinality is a measure of the elements of a set. The cardinality of the infinite set of natural numbers is aleph-0. An infinite set has cardinality aleph-0 if it can be put in one to one correspondence with the naturals. http://en.wikipedia.org/wiki/Bijection This counterintuitively includes the rational numbers also. http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
The real numbers can't be put into one-to-one correspondence with the rational numbers. So the infinite of the reals truly is bigger than the infinite of the rational numbers.
Also see this ppt: http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15251/di...
Regarding calc, I had a strange experience where I kept insisting calculus only logically works if it is based on infinitesimals, which I thought was obvious, but the people I was talking with insisted that it was based on limits. They couldn't understand that an infinitely small value has to be more than zero if an infinity of them is to add up to anything more than zero.
In general, my math education seems to have been particularly bad at dealing with infinity.