I agree with some answers, but will phrase it a bit differently.
The theorems are "what". The proofs are "how" and "why".
But not even this is the full beauty of maths. You start with some formal definitions, the axioms. Basically, you let things be like this, assume that in a new, empty, from-scratch world things go like that. From those axioms you can conclude some further statements, the theorems. You can prove them, basing on the properties of the prior facts. (As I often tend to say, the art of a mathematical proof is to obtain a stronger statement by chaining weaker forms of already known statements.) Then, repeat, repeat, repeat. You are building a beautiful building from reasoning on top of the axioms you chose.
And now, the crux. There is not a single set of axioms, you can use multiple. Of course, many would use the already existing sets, as it suffices in that field. Things like the euclidean axioms of geometry on a plane. Or ZFC. Or some kind of a logic. Or category theory. But you can totally switch to a different set of axioms if you need to. Or even a build new one, if you need to.
Next, about proofs. There are proofs that basically show, that if the theorem in question does not hold, then the world is broken. Those are the contradiction proofs. But there are also proofs, that are constructive. Basically, a theorem merely states "foobar exists". A constructive proof would show you, how to build a foobar (whatever it is).
Proofs can be tedious and exhausting, which is why they are often excluded or only glossed about in the typical school setting. I daresay, if a child is interested enough in the subject, then using the typical childish curiosity, "and why is that?", is a possible way to tackle proofs. "It is because of this, but you might grow tired from all the details."
Next issue, the proofs are typically written down to be watertight. To convey all the needed information in all the detail. (Later, they might miss out some steps, but leave enough to connect the dots, more on it below.) One thing, which is in my opinion just as essential as a correct proof, is the proof idea. A rather short and a more approachable summary of how the proof works. ("Square root of two is irrational, because if we'd assume it is not, then it's a reduced fraction. But we would be able to reduce it once more, which is a contradiction")
Now, for connecting dots. During my studies quite some time ago, I stumbled upon a formal definition of what is "can easily be seen", "is trivial" and the sort. It basically means, the intended reader should be able to figure it out in 15 minutes with a paper and a pencil. Which is a lot for an unprepared mind. Basically, you'd need to build up some kind of a frustration tolerance in a young mathematician, ensure they don't think of themselves as dumb and unworthy if they cannot figure a step in the proof, if they cannot connect the dots immediately. Asking around helps. Oral explanations tend to go in the direction of the "proof idea" and hence help with the understanding a lot. (Which is why lectures are important.)
To summarize: Theorems are the "what", proofs are the "why". One of the best elementary examples of this way of thinking is classical plane geometry, the things Ancient Greeks knew. Geometry is much closer to "real mathematics" than any other pre-university math-related subject. It takes some skill to read the proofs, to extract the "idea" from it. Aspiring mathematicians need to be wary of trying to gasp everything in this very instant and without help. Some kind of explanations on your side or video lectures might also be a great help.