Sure, but cameras don't take pictures instantly. A CCD chip, for example, integrates an exposure over a finite time interval. A typical shutter speed is something like 1/60th of a second I believe.
So if at the beginning of the exposure, the subject is on the right hand side of the field of view and is moving to the left, the light the subject emits will initially be dumping photons on the right hand side of the CCD chip. However, as the subject moves, photons will begin dumping themselves towards the left hand side. This creates a blurry image.
This concept is VERY apparent in astronomy and is why telescopes must do what is called 'tracking' in order to have imagines that are worth something. The dimmer the object you want to see, the longer the exposure. So if you want to see an object that is so dim that it requires a 10 minute exposure time (very common!), you have to have the telescope track objects in the sky so that it rotates with the rotation of the Earth. If you just let the telescope sit there, over the 10 minutes the star in question will move along the field of view and instead of depositing photons in a small, concentrated area of the chip (giving a crisp, clean image), it'll trail off as the Earth rotates, dumping photons in a wider area making it very blurry.