This is a really good question. As any object travels through the atmosphere, it produces air pressure waves (just like a boat travelling through the ocean produces waves). When the speed of the object surpasses the speed of sound, these air pressure waves coalesce, forming a shock wave which “travels forward from the generation or ‘release point'”. If the object maintains supersonic speeds, it is constantly releasing these shock waves. The sonic boom which we hear is therefore the sound of the building up and releasing of the pressure these shock waves create, known as the “peak overpressure”.
For a video explanation of the same, see here:
A very common analogy is that of boats creating wakes. When a boat moves slowly, the water makes waves in front of the boat that the boat flows through, but when the boat goes faster than the waves, it creates a V shape in the water with the point of the V at the nose of a boat, called a wake. In the air, sound moves in waves ahead of the plane, but when the plane goes faster than sound (about 700mph), the sound waves build up and combine, creating the sonic boom you can see and hear. Just like being on the lake after a boat goes by and the wake reaches a previously calm shore.
A sonic boom is in many ways comparable to the wake on a boat. Just as a boat creates a wake of waves, with a larger wake occurring with a faster boat, so too does a plane create a wake of sound when it flies past. When the plane flies faster than the speed of sound, the sound waves produced travel slower than the plane, and therefore become more concentrated of a wake. The resulting compound of sound waves creates a sonic boom, which is basically just a bunch of sound waves that are very close together, and occur suddenly (in your perception).
The “sonic boom” noise is essentially the product of extreme air pressure around the air craft, bullet, etc in question. As an object moves through air, it creates sound waves (vibrating air) in front and behind it, much like the way water is disturbed around a moving boat. At mach 1 (the sound barrier, or speed of sound) these waves become so tightly packed that they create one “super wave” that emits the defining “sonic boom” sound.
You could think of it like the 5 rings from the “Captain Planet” cartoon. When the power all five rings (earth, wind, fire, water, and heart) were combined, they released the power of Captain Planet. The sound waves moving around a super sonic airplane combine in an analogous way to create one “super sound” that is much louder than the individual waves would be separately.
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