Even the closest asteroids are so far away that they look like "dots" in the night sky - just like stars! So, you might think it is very hard to spot them, even with a big telescope.
But unlike stars, asteroids are part of our Solar System and orbit the Sun. This means we can spot them moving against the stars which seem not to move in the background.
You can see this in the short animation on this page. Here, 4 observations are shown one after the other (we call this "blinking" the images). The stars do not move, so it is very obvious which object is the asteroid.
And that's all there is to it. Get several images, use the image software to open them in order, "blink" them, and if you spot something moving across the sky, it's an asteroid!
Common questions:
Why do the stars in the image look like they are moving as well?
Stars can appear to "wobble" slightly due to wind hitting the telescope during the observations, or slight changes in the position the telescope is pointing between images, but asteroid movement is more obvious.
Why is my image just a black square?
You may not have scaled your image enough.
Why does my asteroid jump back and forth?
You may have opened the images in the wrong order