You press one brake when you want to help the tractor turn. You press both when you need brakes to help stop after pressing the clutch in. Locking the brakes together is a good idea when traveling at top speed so you don't accidentally press only one and spin the tractor around. But locking the brakes is a good idea only if the two brakes are adjusted evenly, which may not be the case with a tractor the age of yours. If they aren't adjusted evenly locking them means you are pressing one harder than the other, which will try to make the tractor turn dangerously.
The transmission is exactly like the one in a car, if the car is over 70 years old. Cars of that age, and tractors much more recently, did not have synchronized shifting, which means you, not gadgets in the transmission, have to make the gears turn at the right speed in order to engage them. In a tractor, the easiest way to do that is to stop. In a car or truck you had to learn to do it with some skill.
Terry