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Why do planes break the sound barrier?
Short answer
A plane breaks the sound barrier when it exceeds the speed at which the pressure waves it itself generates propagate, about 340 m/s (1,225 km/h) at sea level and less at altitude. On reaching it, the waves pile up into shock fronts that spike drag; it takes ample thrust and a design with swept or delta wings and a slender fuselage.

What the sound barrier is
An aircraft in flight disturbs the air around it, and those disturbances propagate as pressure waves at the speed of sound: about 340 m/s (1,225 km/h) at sea level and 15 °C, and around 295 m/s (1,060 km/h) in the stratosphere, where the air is colder. As long as the plane flies slower than its own waves, those waves race ahead of it and the air up front "knows" it's coming and begins to move smoothly out of the way. When the plane approaches that speed, Mach 1, the waves can no longer escape forward: they pile up into an abrupt compression front, the shock wave, and the behavior of the air around the aircraft changes radically.
What happens in the transonic zone
Before the whole aircraft reaches Mach 1, the accelerated air over the most curved part of the wing already becomes locally supersonic. Shock waves appear over the upper surface that detach the boundary layer, drag multiplies and the controls lose effectiveness or even reverse their effect. This zone, roughly between Mach 0.8 and 1.2, is what tore apart several aircraft in the 1940s and gave rise to the idea of an impassable wall. The Bell X-1, piloted by Chuck Yeager, proved in October 1947 that no such wall existed: with a fuselage inspired by a bullet, a fully movable horizontal stabilizer and a rocket engine, it passed through Mach 1 in level flight.
The design that makes it possible
Flying faster than sound requires three things. First, abundant thrust: wave drag is so great that many fighters only overcome it with afterburner, though some modern designs reach supercruise without it. Second, shapes that delay and smooth the shock waves: swept or delta wings, thin airfoils, pointed noses and the area rule, which requires the aircraft's total cross-sectional area to vary smoothly from nose to tail. Third, controls and structure capable of withstanding the shifts in center of pressure and the loads of the supersonic regime.
The boom and its consequences
The shock waves that accompany the aircraft spread to the ground like a cone and are perceived as a double boom, one from the nose and another from the tail. That noise is why Concorde could only fly supersonic over the sea, and why fuselages are now being researched that reduce the boom to a muffled thump, like NASA's X-59 demonstrator.
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