Threshold refers to the point at which a dog moves from being able to think and respond to cues, into a reactive or overwhelmed state where they can no longer take in new information or respond calmly. A dog who is "under threshold" is coping with whatever is around them. A dog who is "over threshold" is not.
The idea is used constantly in behaviour work, particularly with reactive or anxious dogs, because training only works effectively while a dog is under threshold.