Keeping it short? Viruses reproduce by inserting DNA into our cells' nuclei (Even the RNA viruses - there's just extra steps).

Several viruses have a dormant stage, where they infect a cell, but don't produce new viral particles unless triggered - usually by stress of the host. Herpes simplex is a good example, which is why cold sores come back.

Once a viral gene is in a stem cell, it's up for reuse and adaptation via evolution, just like any other gene.