How is it possible to perform a rapid climb / tight loop without stalling?
Short answer:
The maximum angle of attack is never reached in this video, thus the aircraft is not stalled.
Longer answer:
Stall is a problem mainly occurring during low speed flight. Our angle of attack is always depending on our flight path.
Let's first assume, that are at cruising altitude in a level flight condition. In this case the pitch angle of the airplane equals our angle of attack. As generated lift is dependent on your airspeed and the lift coefficient (which is again dependent on your angle of attack), reducing airspeed in level flight while maintaining your altitude will force you to in increase the angle of attack. At some point a further increase will result in a too high angle of attack and thus stall the aircraft.
Now, let's look at your problem:

(source: aeroskytech.com)
The main difference is your speed, or so to say the excess power used for climbing. During climb, the trajectory of the plane is not equal the horizontal axis. Therefore, also your angle of attack is not equal to your pitch angle (the angle between the longitudinal axis of the plane an the horizontal axis), but to the angle between the trajectory and the longitudinal axis. During the loops showed in the video the airplane is not only changing its pitch angle but also its flight path, therefore it is not stalled.
An interesting example for maintaining your flight path while increasing are military aircraft in combat maneuvers, their pilots rapidly change their pitch angles while still flying in the same direction. This works as a decent speedbrake, allowing them to intercept other aircraft.