'Betcha' is a contracted form of 'bet you'. I understand that 'you bet you' doesn't seem to make sense, but it is an informal expression afterall. Another relatively common one is 'dontcha' which is 'don't you' contracted. They are quite 'American' contractions - I don't hear them much outside songs/films/tv programmes from the US.
I've always thought that it was a contraction of "you (can) bet your (life on it)".
I did some searching and Urban Dictionary gives this explanation. It also doesn't make much sense alone However, it at least explains how it could come to life. You-bet-you's etymology is vague to me. Thank you for your answers.