Simple: Running water helps flush food particles through the garbage disposal, into the trap and down your waste pipes without gumming up the moving parts.
It’s like swallowing an vitamin with a mouthful of water. Can you do it without? Sure, but it makes it easier.
The one exception to this rule is liquid grease or fat. Water, especially cold water can make these items congeal, and they will likely solidify in the trap or nearby, creating a potential bottleneck. Instead, use hot water and a little bit of dish soap. The heat and soap break up the grease, and will at least spread the congealed solid into a thinner layer in your pipes. If you are thinking to yourself that anything that results in a “congealed slime like solid” shouldn’t go into your pipes, you’re right! You’re better off by disposing fat and grease into the trash. So remember to turn on your water before turning on the disposal.