What are coins for in Super Mario Kart on SNES?

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.

As michaelb958 notes, you only need 10 coins for the maximum speed bonus. Any additional coins will not provide further benefits.

You will lose 4 coins if are hit by an item that causes you to spin out, 2 coins if Lakitu rescues you, and 1 coin if you hit another player. If you crash into another driver while you have 0 coins, your kart will spin out.

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.