I'm not sure how much work has gone into investigating this, but there is apparently 2 reasons to collect coins. The first is that the more coins you have, the higher your top speed. So if you and another player with the exact same kart had a drag race, a player with 10 coins could reach a higher top speed. This is obviously only useful if you are reaching the top speed. The second is that you are meant to get a small speed boost when you pick up a coin. It isn't huge, but you are meant to get a little kick for each coin, even if you have collected the max of 10 coins already.