All perching birds have a reflex action in their feet that causes tendons in thier legs and feet that makes them grip the perch firmly until they conciously relax the leg tendons. We've all seen videos of hunting eagles snatching fish out of the lake holding the fish firmly... same process... Addiitonally, the perching chicken rests the weight of its body on their breast against the perch... all very secure...
The sinew that closes the claw runs over the back of the knobbly bit (actually the heel) and is tensioned when the joint is tightened such as when they sit down. If fact, a perching bird can not release its grip until it stands up and releases the tension.
Anyone who's had their grandad kill a chicken for sunday roast and let the kids play with the legs will tell you that these sinews still work even when the leg is removed from the chicken. Chasing your sister with a leg while tugging the tendon so making the claw open/close and running behind your sister scaring the **** out of her is not a nice thing to do.
I am afraid the fishing bit with eagles is a bit of a fairy story. A bird of prey will carry its catch head first into the wind to minimise resistance when flying so can not tug its legs up and gain advantage of its sinew action.