The reason is historical.
Fixed gear bikes have been traditionally used for track racing, long after most other bikes had switched to using freewheels.
In the high-speed velodrome environment, it would be dangerous if the rider in front of you could brake suddenly: you would crash into them, and likely most of the group behind you would join in the fun
(this is also true in a paceline or peloton on normal roads).
So brakes have always been banned, and aren't really needed on the track anyway.
But track bikes sometimes get retired to road use.
And as fixed gear has gotten more popular, many frame makers are building similar bikes for the road.
Some of them are closer to track bikes (no brakes), others might include front brakes or both front and rear.
bicycles.stackexchange.com/users/157/darkcanuck
You surely must mean a fixed gear bike.
A freewheeling bike with no brakes is going to require you to stop via crashing, some sort of flinstones-esque maneuver, or a ted shred move.
A fixed gear bike can be ridden marginally safely because the pedals are locked to the rear wheel in both directions, so locking up your legs can slow down the rear wheel.
People do that because they believe it gives them a better connection and feel with the road, a more 'zen' riding experience, and that the bike has better looks.
Fixed gear bikes are ridden in track racing with no brakes, and so the track bike origin is part of why people ride them on the street with no brakes.
There are still many people who have brakes on fixed gear bikes.
I have a fixed gear, but I have brakes on it because I prefer to stop quickly and safely. People that ride coasting bikes with no brakes are insane.
Lastly, this excludes coaster braked bikes, which have a brake built in the hub, common on bikes for small children.
bicycles.stackexchange.com/users/683/whatsisname
A track bike has no brakes because braking would cause a pile up.
When a fixie on the road has no brakes it is because someone is being macho.
They will insist they can stop just as quickly or they can always see the road and path ahead.
The former being easily disproven and the latter generally being optimistic.
There seems to be a fair overlap with the same crowd that shuns helmets in traffic.
I ride a fixie with two brakes (rear rarely used).
The bike store near me sells fixed gear bikes set up for both track (no brakes, swept handlebars so as not to catch) and for road (with brakes and regular handlebars).
bicycles.stackexchange.com/users/389/LanceH
My fixed gear bike has brakes.
And I wear a helmet when riding it.
And I don't listen to my iPod while on the bike.
I'm the new non-conformist.
bicycles.stackexchange.com/users/638/alesplin
Why spoil the clean look of a lumberjack shirt, rolled up jeans and espadrilles with hideous cabling, calipers and pads?
bicycles.stackexchange.com/users/715/Paul Gordon
I'm not sure about that assumption, while you can ride a fixed without brakes and use solely pedal pressure, I wouldn't want to rely on it on the road.
Track bikes don't have brakes and historically fixies came from the track.
My fixed has both front and rear (although I don't use the rear much as it chews through pads) and while I can brake reasonably well with just the pedals, through in the front brake too
and I can stop on a sixpence.
Of course, in the UK to be strictly road-legal you must have at least one brake.
bicycles.stackexchange.com/users/425/Unsliced
It seems to be assumed you will not have any brakes on a fixed gear bike, why?
Why don’t fixed gear bikes have brakes?
bicycles.stackexchange.com
<brakes><fixed-gear>
bicycles.stackexchange.com/questions/2002
bicycles.stackexchange.com/users/132/Ian
Track bikes don't have brakes and historically fixies came from the track. They have become more and more popular and are now made for the road. Their track origin is part of the reason why people ride them on the road with no brakes. When a fixie on the road has no brakes it is because someone is being macho.   Historically, fixed gear bikes are used in track racing, where breaks are unnecessary and could in fact create a pile-up crash. Retired track bikes, or bikes inspired from this design have followed the same tradition. Regardless, fixed gear bikes are often a stylistic choice, as they arguably give a better look to the bike and driver alike, and allow for a more natural riding experience.
Historically, fixed gear bikes are used in track racing, where breaks are unnecessary and could in fact create a pile-up crash. Retired track bikes, or bikes inspired from this design have followed the same tradition. Fixed gear bikes are often a stylistic choice, as they arguably give a better look to the bike and driver alike, and allow for a more natural riding experience.