The lysogenic phase of a viral infection is when the virus injects its DNA into the cell. Instead of the DNA being replicated and then formed into new viruses (lytic), the virus' genome is incorporated into the HOST DNA genome by Recombination.
Essentially the viral DNA inserts itself to the host genome, also has all of its mRNA protein coding regions deactivated to prevent it damaging the host. Think of it like a dormancy stage where it divides along with the host