I think what people are forgetting in their rush to yell "a cadet taking over a ship?!?!" is that James Kirk is a magnetic leader.
My guess is that the rest of the bridge crew noticed Spock wasn't totally in control, but were unsure how to react.
Then Kirk takes command of the situation, and thereby establishes himself as the leader of that group.
Alexander the Great was leading armies at a very young age, in command of people with much more experience who were much older. Leadership can be an innate quality, and we've already seen that Kirk has it.
As the older Spock tells him, it's time for him to take command of HIS ship.