People inside the plane didn't hear a thing. In fact, the transition to supersonic flight was so unimpressive that they decided to install a “mach meter” inside the passenger cabin so that people would know then they passed the speed of sound.

Contrary to popular belief, passengers inside the Concorde could not hear the sonic boom as the aircraft broke the sound barrier (Mach 1). Because the plane was traveling faster than the sound waves it produced, the "boom" was left entirely in its wake, trailing behind the aircraft on the ground. Inside the cabin, the transition to supersonic flight was remarkably smooth. The only way passengers knew they had crossed the threshold was by watching the digital Mach Meter on the cabin bulkhead, which would tick up from Mach 0.99 to Mach 1.00. While they didn't hear a boom, passengers did experience a slight "shove" in their seats during the transition, and the sound of the engines would actually become quieter at supersonic speeds because the noise was being "left behind." To people on the ground, however, the Concorde produced a thunderous "double-bang" that eventually led to it being restricted to supersonic flight only over open oceans.