Suppose that you want to teach the 'cat' concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying 'kitty' each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience.
Only professional mathematicians learn anything from proofs. Other people learn from explanations.
Somebody came up to me after a talk I had given, and say, "You make mathematics seem like fun." I was inspired to reply, "If it isn't fun, why do it?"
