Statistics and demographics.

Candidate A and party A tend to get X% in demographics Y and Z. Candidate A won precincts 1, 2 and 3 by B% last time, and did 2% better this year. That lets them predict several other precincts, and when enough precincts can let them see enough trends, they can make pretty reliable predictions. It doesn't work when things are too close, so the closer it is the longer it takes to call.

Sometimes no one calls a race that night, sometimes they call a race at the moment the polls close.