This usage of the word "tap" has nothing to do with tapping on a door. It is a phrase that uses a different definition of "tap."
Originally, "tap" could be used as a noun meaning a plug for a hole. In a barrel of beer, for instance, the person opening the barrel would make a hole to let the beer out. The hole would then need to be plugged and it would be plugged by a "tap."
From there came the use of "tap" as a verb. It meant to take the "tap" out of the barrel and let the contents flow out. That led to the figurative use of "tap" that you are asking about. "Tap" or "tap into" draws on another meaning of "tap," to reach into or open up. "Tap into" means to have access to. "On tap" has come to mean "to draw from" or "to draw upon."
So, in the sentence you give us, saying familiar things accesses or activates a different part of the brain than spontaneous speech does.
Random House Dictionary:
tap: to penetrate, open up, reach into
Verb phrase "tap into" Informal: to gain access to