Imagine a boat sitting in the water creeping forward at a very slow rate. It will send a ripple of water out in front of it, albeit a slowly moving one. You've got to imagine the boat moving ar around a couple of centimeters per second for this. But if the boat picks up any speed, it crashed into the water in front of it faster than the ripples can escape. It no longer creates a swell of water in front, all the water-wave ends up trailing out behind it in a V shape, what's called the "wake" of the boat. If you watch the boat going past you, the water triangle will follow soon after and splash you. This is an aquatic boom. A sonic boom works the same way except with air instead of water.
As a plane moves through the air, it smashes into air particles and scatters them out in all directions, including in front of it. The speed of sound is, on average, about 333 m/s (varying slightly with temperature and altitude). If the plane is travelling at 100 m/s, then the air particles will go flying away, leaving the plane behind. But let's say the plane is moving at 400 m/s. At this point the plane is bumping into air particles, but also catching up before they escape. The air-build up in front of the nose can't get away.
The air will end up with lots of movement energy (kinetic energy) but it can't move forwards. Instead, the air ends up forming a cone around and behind the plane, trailing behind it like ripples on the lake behind a boat. The plane is moving so fast it drags a vortex of air along with it. When this enormous cone of air hits your eardrums, you hear the sonic boom.
This image demonstrates it beautifull: