Perhaps I should backtrack for a second...
In the wild, birds wear down their beaks and talons naturally, just going about their daily routine. In many cases, zoo birds do to. Sometimes, however, they get a bit overgrown and need a bit of extra TLC from the staff. Typically, birds in large aviaries with lots of varied perching and incentive to fly about wear their claws down just fine. Birds wear down their beaks when feeding on hard food items (scraping against bones, or working on hard-skinned fruits or nuts), but those fed softer diets may not do as much natural maintenance.
All of this means keepers must trim the offending beaks and talons, or face the problems of severe overgrowth. For example, too-long talons may curve inward and poke the birds in the sole of the foot, causing a wound (what keepers often call a bumble).
Trimming is often done with a Dremel or similar power tool, used on a low setting so as to not accidentally injure or rattle the birds. The claws and beaks themselves are made of keratin, the same substance as our hair or fingernails, so the act of trimming them does no damage to the birds themselves. Done by a skilled handler, the process takes just a minute or two.
I'm only semi-skilled. Despite several years of doing this, I still don't have the best eye for what the beak or claws are supposed to look like, resulting in lots of stopping, looking, judging, grumbling, and resuming. Combine that with a handler who doesn't always maintain the best grip, and sometimes things get a little hairy... or feathery.
In an effort to make the process less stressful for birds, many keepers have begun training their charges to allow for voluntary nail trims. The bird is perched and offered treats, while the keeper is able to lift one foot, then the other, and gently file the nails down. A trained behavior means that it can be performed more often (without the hassle of netting and grabbing), which in turn means that the job typically takes less time to perform.