I only have two small things to add to Shaun's excellent description (there were some things in there I hadn't heard before. I want to check them out with some folks who do some different styles to see if they have these elements in common).
First, in the strictly physical sense the action of shifting the weight forward in this exercise is actually one half of a "centered" walking motion. I call it the "power walk" when I do workshops for non-aikido folks. The shifting of the weight forward allows you to set up a solid one point base with the forward foot so that you can move the back foot into the forward position, then you shift forward again and repeat the process. This is actually how we move in aikido if we are connected to the earth and have power even when we are moving. This is the first step to getting your body to do a technique, not your arms.
Second, if you get the proper motion down in this exercise, a partner can grab both your wrists and try to provide resistence. You should be able to stay relaxed yet have tremendous power in the movement. You can actually lift the person off his feet with the extension and totally break their balance on the draw. Also, Mary Heiny Sensei once told me that the extension movement should be done explosively with the energy of a strike. One of the things I have done when teaching the exercise is to have someone hold a focus mitt on their chest and the students do the exercise and actually hit the mitt with the back of the wrist on the extension. The effect is amazing in that, with a completely relaxed movement, you can knock someone back several feet. Anyway, these were basically things you could do to check whether your "rowing" exercise was really coming from your hara or not.