In Khan Academy's RSA Encryption 2 video, I can't understand how they have got from
$$m^e\bmod N\equiv c$$ to $$c^d\bmod N\equiv m\,.$$
The transcript says,
01:39 some piece of information that makes it
01:41 easy to reverse the encryption
01:44 we need to raise $c$ to some other exponent, say $d$,
01:48 which will undo the original operation applied to $m$
01:51 and return the original message $m$.
01:54 So both operations together, is the same as
01:57 $m^e$ all raised to the power of $d$,
02:01 which is the same as $m^{e\times d}$.
The person says if we want to reverse the function to give us $m$ (the encrypted message), we simply have to raise $d$ to another exponent, $d$, and this reverses it.
Although I know this video isn't supposed to fully explain everything I don't understand how this is done. How can you just raise $c$ to some power and get the original message $m$. Is there some sort of proof or can someone explain it in more detail perhaps? By the way, I pretty much know nothing about encryption/ cryptography. I study CS at college (UK) AND was just watching due to curiousity.