Falcons wear hoods to keep them calm and ensure they are alert when the falconer needs it. A falcons sight is up to 10 times better than us humans. This means the bird of prey's visual stimulus and input is very high and it can see many potential prey targets long before us humans can. This means that a falconer may see a target and release the bird, however the bird has long seen something else that has it's fancy and goes it's own way after it's own target, or worse yet it is so frustrated at seeing so many target come and go, it just goes not want to go after prey anymore. The falconer does not want this and keeps the falcon hooded until he sees the target he wants, then removes the hood and the close (falconers) target will immediately catch the falcons eye putting it into hunting mode with the falconers chosen target as prey. The falcons are also trained to go into hunting mode once the hood is removed.
A good falcon hood does not bother the bird of prey at all. It fits well, does not damage feathers or hamper breathing allows air circulation and does not under any circumstances come into contact with the falcons eyes.
This manner of working with falcons and eagles was learned many hundreds of years ago already. Which has lead to a rich falconry heritage in the art of hood making from all over the world.