Exonucleases function by cleaving nucleotides one at a time from the end of a polynucleotide chain. They can be classified based on their directionality: 3' to 5' exonucleases remove nucleotides from the 3' end, while 5' to 3' exonucleases remove them from the 5' end. The specificity and activity of these enzymes are vital for maintaining the integrity of genetic information.