As I understand it, the purpose of the TSA dog is to detect explosives. It was communicated to me that dog is constrained resource that would somehow alleviate a bottleneck. How would having an unlimited supply of TSA dogs alleviate long wait-times?
Generally, a handler will walk with the dog along a line of luggage or passengers. The dog will 'alert' if it smells something it's be trained to notice like chemicals used in explosives or food. An alert can be a bark or the dog sits down, something like that.
I've seen dogs working at baggage claim and at lines for security screening.