Why do cyclists have their seats so high?

The KOOTU Cycling Team summarizes the two main reasons, that boil down into; cyclists that don’t have their seats “so high”, probably have them set too low.

Seat Height in relation to the pedals, or ground. If you are a novice cyclist your seat probably needs to be higher than you think, in order to make your cycling more efficient. Many people think that you need to have your seat set low enough that both feet can be firmly on the ground when we stop. We teach kids who are just learning this sometimes. For reasons of safety, stability and confidence. Then it can stick with people their whole lives.You don’t engage the large leg muscles (thighs and glutes) until your leg is nearly fully extended, these are the muscles that make cycling much easier. If you have your seat set low, and your knee is still very bent at the bottom of the pedal stroke, you are cycling mostly with your calves and only a little thigh muscle.
kootu bike fit
The broad advice for a proper seat height, that allows efficient use of the larger leg muscles, and therefore more efficient and easier cycling. Is that with someone holding the bike upright for you, your seat height should mean that your heels just firmly rest on the pedals with your leg straight. Now you should pedal with the ball of your foot. So this seat height will mean that your knee will still have a slight bend (and so not be over extended) at the bottom of the pedal stroke.
kootu aero bike
This will result in a seat height that is higher than most novices are used to, or possibly comfortable with. You have to get used to the idea that it is perfectly safe. You just have to get used to putting only one foot down and tilt the bike to that side when you stop. Or side forward off the seat to put both feet down.
KOOTU Bicycle
Seat height in relation to bar height. The biggest limiter to going as fast as you can are gears (another story) and wind resistance. The biggest factor in wind resistance is the rider. If you want to go fast you have to lean right over to make an aerodynamic (aero) shape, rather than sit upright like a big wind catching sail. To achieve this race bikes are set up with a bar position that is low compared to the seat. Often by a good few cms. The seat is the same height from the pedals or ground as other bikes, but the bar position makes the seat height look much higher. This is an example from u/brovnic on reddit. The seat isn’t particularly high, It’s the bars that are low.