The security of Zero knowledge proof performance
I don't know the term "safety". But when it's written 'the process is more secure'. It means that the probability for an adversary that doesn't know the secret (here how to go through the door) decreases "exponentially".

For example, to solve 4 challenges the adversary has to choose the side door $4$ times, so it has the probability $\frac{1}{2^4}$ to win the game.