Why do moving subjects look blurry in photos?

First, I'll talk about what cameras do normally, then about how motion affects this operation.

In order for an image to be sharp and in focus, all light coming from a single point on the object being photographed must fall on a single point on the film or sensor. If you take a picture of a face, you want all of the light reflecting off the left eye fall on one part of the image sensor and all the light reflecting off the nose fall on a different part. If the picture is out of focus, the light from different parts of the face can hit the same bit of sensor, and light from the same part of the face can be spread all over other parts. This results in a picture where every part of the face is mixed up with the other parts. This is called a blurry image.

If the subject is moving, a similar blurring occurs because the shutter of a camera is open for a span of time. Imagine you're taking a picture of a person, and that person moves their hand. When the shutter first opens, the camera directs the light from the person's had to a certain part of the image sensor. However, because the hand is moving, light from the hand's new position will be directed by the camera to a different part of the sensor. So, the camera will receive light from all of the positions of the hand while the shutter is open. That light from different hand positions will end up on different parts of the sensor. This results in what looks like a smeared image of a hand tracing the path of motion.