In the oldschool super mario kart for snes you could collect coins that lie around on the track.
What do they do? Why should I collect them?
Back in the days as a kid I would collect them because they are a shiny. I doubt that is everything to it.
Each coin held slightly increases your top speed on the road, by approximately 0.8%-1% per coin (the exact percentage depends on the selected driver, the kart class, and whether you're playing a PAL or NTSC version). For example, a Koopa Troopa with 10 coins will have the same top speed as Bowser with 0 coins.
Coins do not affect any other aspects, so drivers are still going to vary a lot in acceleration, handling, etc.
Also, when a player-controlled driver is hit, they lose a coin. If a player-controlled driver is hit with zero coins, they will spin-out.