There is a little dance that goes on that isn't in the clip. Someone calls in with a request to land and there is traffic preceding. Example:
Tower: You're number 2 following Cessna traffic on a right downwind.
Acft: Number 2, looking.
...
Acft: Traffic in sight.
Tower: Cleared to land number 2.
If you're given a sequence number, it can only be given after you report the traffic that you are supposed to follow in sight, and then you are cleared to land only when that traffic has landed and fully exited the runway. The controller will arrange things so that there is enough time for that to happen. If for some reason it doesn't you will go around.
In this example, the pattern is fairly busy and the confusion aircraft ends up fourth in line to land.