Freezing the wart irritates the area, causing a small amount of swelling, redness, and sometimes pain at the wart site.
The area is injured by the freezing, and in that inflamed, injured area the immune system responds by increasing its presence in and around the swollen areas. Like a city adding more cops to a area with a spike in crime. The hope is that the increased immune surveillance, one of the immune cells will find on of your skin cells infected with the wart causing virus (HPV) and sound the alarm to your immune system to begin fighting HPV infected cells.
Why doesn't your immune system do this already? One, your outer skin, like most areas with little circulation, gets little immune surveillance. Your immune cells, like cops, need roads to get to the site of a crime/infection. You can increase the surveillance by causing a small amount of damage. The other reason it may take a few treatments, is that HPV is very good at hiding inside your outer skin cells, so your immune system may not pick up on it on the first pass.