Why doesn't the world blur and jump every time your eyes move?

Because your brain switches off the blurry part. Your eyes don't glide smoothly across a scene. They jump between targets in quick movements called saccades, and during each jump the brain selectively blocks visual processing, so neither the motion blur nor the gap in perception is noticeable. The seams are edited out before you ever see them.

The jumps are fast and constant. A saccade can reach a peak speed of up to 700 degrees per second in humans, and the eyes move as fast as they are able, because the speed cannot be controlled. A typical reading saccade lasts only 20 to 30 milliseconds. When you look around, you make two to three fixations a second. The reason is that saccades let the eyes sense small parts of the scene in greater detail, and the brain assembles the pieces into one continuous picture.

The brain could not simply watch the blur. The visual system is too slow to process images slipping across the retina at more than a few degrees per second. So in every jump it cuts the feed instead. This is called saccadic masking, and it was described in 1898 by Erdmann and Dodge, when they noticed that an observer could never see the motion of their own eyes.

You can try it. Stand at a mirror and look from one eye to the other. You will not see your eyes move, yet a friend watching you would see them dart. The brain hides the movement, and it hides the evidence that anything was hidden.

There is one strange side effect: the masking is linked to the stopped-clock illusion, called chronostasis.