The trainer waits for the dog to stop barking, then reinforces the silence. To speed up the process, the trainer sets up a scenario where a person will pass the house several times. He reinforces the dog heavily in the moments of silence, ignoring the barking. The dog gradually learns that it is more reinforcing to be quiet, so the barking extinguishes. In addition to extinction, clicker trainers use negative punishment.
Negative punishment is the removal of something in order to decrease the frequency of a behavior.
When clicker trainers want to suppress an undesired behavior, they rely more heavily on negative punishment than positive punishment because negative punishment is equally effective and causes neither fear nor pain. Although clicker trainers try to rely most heavily on positive reinforcement, the idea that they use it exclusively is completely false. We train dogs to make it more pleasant for our species to co-exist. A large part of that training is teaching the dog cues to perform certain behaviors. If you can drive a car with a manual transmission, chances are you shift pretty much automatically.