I've never taken a course in Analysis, but my take on why we rationalize the numberator is simply that the trick works. Now that you've seen what to do with this problem you know that you can do it to others similar to it.
(Eventually you'll take the limit of your answer as h goes to 0. Then you'll see why this simplifies the problem rather than make it more difficult.)
-Dan
They only cancel each other out IF YOU CANCEL THEM! That might sound funny, but the point of us inserting the same factor on top and on bottom is to somehow combine/rearrange so that we get something ELSE to cancel...
"FOIL" out the top. You will have "h" remaining. Just h. This cancels with the h factor on bottom.
Then directly substituting h = 0 will give something meaningful, and the answer.