Why do quadcopters use brushless motors?

For the smallest (and cheapest) of multicopters, DC brushed motors are well in use: http://www.rcgroups.com/forums/showthread.php?t=2009496 They are coreless motors¹ and have very little inertia, can accelerate very quickly and are perfectly suited for high frquency PWM (we drive them with up to 32kHz). You can achieve more than 3:1 power:weight ratio on a sub 50g AUW airframe. Driving those motors is a lot simpler than a brushless motor, you basically just need a transistor and a capacitor.

That applies to micro copters, brushless technology didn't catch up in that weight class (yet). Probably just a question of time. The first micro brushless systems are available, but they cost many times more than coreless motors and don't offer more performance (yet).

For larger quads: no question, brushless is king. But for the micro and nano-quads, its awesome to have these cheap motors to crash with confidence ;)

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^{1 - http://www.pacificelectronicscorp.com/dccoreless.html}

Most motors on quadcopters are outrunners. In outrunner motors, the rotating part is on the outside, and not the inside (as opposed to the inrunner motors). Because of this layout this type of motors can generate much more torque. High torque is required for quadcopters, since you balance by changing the revolutions of the motor. The higher your torque the faster you can change the speed of your propellers. High torque also means you don't need a gearbox, and save a lot of mass. Outrunners are not very practical for brushed commutation, as it would require lots of wires and additional contacts. Hence most of the outrunners are brushless.

The brushless motors are way more powerful for their weight than available brushed motors, and they last longer.

Power to weight ratio is king, in an aircraft.