Basically p is a statement ("honk") and ~p is the denial of that statement ("you're not about to honk"). p ≡ ~p would mean "honk iff(if and only if) you're not about to honk", as Obres put it.

Now we assign a value to p. It can be either true(1) or false(0). ~p is the opposite of the value that we assigned to p, so when p = 1 we have ~p = 0, and when p = 0 we have ~p = 1.

And last, we evaluate p ≡ ~p. The '≡' sign is only true(1) when the left statement has the same value as the right statement (both are false or both are true). See this for an explanation.

So, since '≡' yields 1 only when p is the same as ~p, p ≡ ~p will never yield a 1 because p will never be the same as ~p. Hence the contradiction.

Sorry if I screwed up some terms, I'm really rusty on my formal logic.