One way to explain OTPs is a proof by induction.

The base step:

Start with one bit. The key is also a bit chosen at random, and the cipher text is plain XOR key.

You can then work out that no matter what key is chosen, you have no reason to believe 0 or 1 is the original plain text.

The inductive step is simply to add a new random bit of key that is independent from the prior key, and a new bit of plaintext.

What I like about this explanation is that you can actually work out the decision trees for a few bits to convince yourself that the encryption still holds.