It is the plot conclusion for the Jake Morrison character. He is a pilot who solves problems by brute force. At the start of the movie he is on the moon moving a large laser cannon into it's docking position, but when something goes wrong he uses the ships engines to fix the problem.
He is punished for his heroism, and during the film we find out he almost got Dylan Hiller killed during a training exercise (that's why he punches him when they meet again).
It's Jake who thinks of using the cold fusion drive to break free of the Queen's control over the hive of ships. He again applies brute force to solve the problem.
The point of the scene is to vindicate him for his past, show that he's really a hero and for some reason prove that brute force is always the better way.
It is one weak plot conclusion for a weak character in a poorly written story. It feels like the writers added this at the last minute, but the fact that it connects back to the beginning of the film implies they had this ending in mind from the beginning.
The only evidence in the film to indicate why they could use the cold fusion drive is when Rain Lao reaches over her head to pull down two large handles. This visual indication implies that the cold fusion engines could be manually fired, without any guidance, any pre-computing of the trajectory and the ships would just fire the engines in full power.
After the engines fire the ships accelerate smashing other ships in their path. The Queen takes notice of this, looks at the two ships and releases her control over them. I guess she releases control to stop them from smashing other ships, but the reason is not given in the film.
Before they fired the engines, Jake Morrison says "every tornado has an eye, right?". He is implying (and incorrectly I might add) that the spinning rotation of all the ships will send them up the center like a tornado would do.
It's all wrong and highly implausible.
- The wind from the spinning ships fails to pick up any debris on the ground. So how could it be strong enough to suck up two ships?
- After the drive is fired. The ships smash other ships in front them without taking any damage. That's like me driving my car into the back of another car, destroying that car without taking damage. Keep in my that the two cars are from the same manufacture and model.
- If you spin a rock on a string the force pulls outwards, but in the film the faster the ships travel the more they move into the center.
- The ships are sucked up the center of the tornado, but they leave a smoke trail behind them that does not move. There is no wind.
- When the ships fall back down the engines have burned out, but they are able to pull out of the dive. This implies that the ships are aerodynamic like an airplane, and will gain lift on their own. Sorry, but they don't have wings and are shaped wrong.
So to answer your question:
Why can they use the cold fusion drive?
Because there are two large handles that immediately fire the engines if you pull them.
Why are they able to set themselves free from the Queen's control?
Because when they fired the engines they started smashing other ships. So she released control which got them sucked up and out the top of the tornado.
Those are the answers the writers want you to swallow.