At first glance, asteroids would seem to be very difficult to spot. Even the closest ones are so far away that they look like "dots" even with a big telescope - just like stars!
However, unlike stars, asteroids are orbiting around the Sun. So, if you take several observations of a bit of the sky, the stars will not move, but any asteroids will.
You can see this in the little movie on the right. Here four observations are shown one after the other (this is called "blinking" the images). The stars do not move, so the asteroid is very obvious.
And that 's all there is to it - get several images, "blink" them and if it moves, it's an asteroid.
Note: that the stars can appear to "wobble" slightly due to wind and pointing variations, but asteroid movement is more obvious.