this is a totally un-rigorous way of looking at it, but rationals are countable b/c we can look at'em this way...
rational = m/n with m,n are integers, n!=0. So for m=1, there are countably many numbers 1/n. For m=2, there are countably many 2/n.
Now you'll just have to take this on faith...but there's a theorem of analysis that says that the infinite union of countable sets (each 1/n, 2/n, 3/n, 4/n....m/n and repeat for negatives is countable) is still countable.
Another way to think about it is that with rationals, we're limited to certain combinations of decimal expansions...irrational numbers fill in all the other combinations (pi, e, sqrt 2, etc). But there is no general expression for irrational numbers, and we cannot write them out in a systematic list the way we can write out 1/n, 2/n, etc.