"How do you find comets?"
It takes dedication, perseverance, attention to detail, lots of hard work - things that these guys have in truckloads to succeed the way they do.
Oh yeah, you need an actual comet as well!
Seriously though: generally it's a matter of taking lots and lots of images of the sky, with repeat images of the same piece of sky at relatively short time intervals, and with enough detail in each image that a comet would show, then by comparing successive images of the same small pieces of sky they look for something, some tiny blurry blob, that appears to move in relation to everything else in the image.