First off, I know almost nothing about driving and do not do it, but it looks fun! I went to a driving schooling show today. I am wondering what is the reason for the person that just sits in the back on some carts? Just along for the ride?
To help the driver control the horse in an accident, by jumping off the cart and getting hold of the horse by the bridle, or helping unhitch the horse in an emergency. Or jumping down to help look for traffic when crossing a road, etc. It's rarely a good idea for the driver to get out of the cart - they have much less control of the horse if they do.
The more horses pulling the vehicle, the more people you need to carry along.
Or to help balance the vehicle going around turns, so it doesn't flip over (if you are going fast or doing sharp turns).
Or if it is a junior driver, they are required to have an adult along at shows, in case of an emergency.