dense means that between any two real numbers, we can find a rational number.
"at most countable" means that there is an injection from our collection, to a subset of the natural numbers (this subset could be finite, or infinite).
what drexel28 is saying, is that A is an "indexing set" for our collection of intervals. this is mostly a notational convenience, so that we can distinguish different intervals from each other while still using a notation that tells us they all belong "to the same collection".
since our intervals are pair-wise disjoint, one possible choice for A is the set of all real numbers {a: a = inf(I), for some interval I in our collection} (so if one of our intervals was (2,4), we would label that interval ).
by your hint, each interval , contains a rational number, which we can call . since the intervals are pair-wise disjoint, no other interval contains this rational number.
but the rationals can be put into a bijection with the natural numbers, let's call the bijection h.
thus if C is our collection of intervals, we have an injection k from C into the natural numbers given by: