Same way your eyes stay fixed on a point while your head moves, and for the same reason.
Bird's eyes are mostly fixed in place by a ring of bone called the sclerotic ring. Since they can't move their eyes, they move their entire head.
Fun bonus: your eyes can twist or tilt around the axis of your pupil. This is why the world seems not to tilt when you tilt your head, up to a certain point which is the limit of tilt.
For people and birds, these are all automatic reactions learned by the brain to hold a stable picture, just like all the dozens of tiny adjustments you make to the muscles of your legs and feet to remain standing still.