How does a polymerase chain reaction relate to DNA fingerprinting?

They are cousins.

The word relate in your question makes me cringe.

There is a specific PCR reaction called AP-PCR (Arbitrarily Primed PCR) which involves taking two samples from different sources or species and running a PCR reaction with the same set of primers. These primers are not too specific (so no annealing will occur) or too general (too much annealing will occur). If the primers anneal and are not too far apart and are facing each other, a characteristic band (fingerprint) is produced. These bands can then be run on a gel along with suspected relatives and the gels can be examined for similarities.