5 Facts

1) Some people think that brakes ruin the whole point of fixies. You don't really need brakes if you know what you're doing but for begginers, brakes are definitely necessary.
2) Braking by resisting the turning cranks greatly increases stress on the knees which can lead to injury so brakes are better for control and your health.
3) You can convert from fixed gear to freewheel by flipping the wheel. On fixed gear, while pedaling backwards, you might unscrew the wheel and wipeout badly. You can put some Loctite on it and it should hold for a bit.
4) There are fixed gear races that require going through alleys and tight corners. The most famous one is the Monstertrack Alleycat in New York.
5) The use of any bike without brakes on public roads is illegal in many places, but the wording is often something along the lines of "...must be equipped with a brake that will enable the person operating the cycle to make the braked wheels skid on dry, level and clean pavement..." which some have argued allows the use of the legs and gears.