Why is probability theory that important to communication theory?
In analog communications, you are trying to recover a message signal from a received signal containing noise. Noise is a random fluctuation of the signal variable. So with out some theory of what random means, we can't even understand what noise is.

Furthermore when we talk about recovering the message signal, we're basically talking about estimating the original signal from the received signal (corrupted by noise). Again, estimation is a fundamental of probability theory.

In digital communications, at the most basic level, we talk about receiving bits of a message after they pass through a channel with some probability of error. Again, this description doesn't even have meaning until we have a theory of probability.