How do you awaken the Ancient One by opening the gate of Kadath?
It's talking about this encounter:

Unknown Kadath
In response to your pitiful knocking, the massive gate of the Castle of the Great Old Ones swings wide. Lose 1 Stamina and 1 Sanity. If you are devoured as a result, the Ancient One immediately awakens.

It's a literal physical gate in the encounter, not a gate like the ones that open normally during the game. And for some reason you decide to knock on it. Pretty hard to awaken the Ancient One that way - you have to manage to get that encounter and already be at 1 Stamina and 1 Sanity.