The human body has two main types of immune defense cells, B cells and T cells. B cells produce antibodies. Antibodies latch on to specific points on a foreign cell, called antigens. The T cells can then move in and destroy the invaders. Over time, some of the activated B cells turn into long-lived memory cells so the body retains the knowledge of how to fight the pathogens, and some will become plasma cells which constantly produce antibodies.
After an infection, or an initial vaccine, the blood is swarming with antibodies. Like cops on a stake out, if there's another infection, they can immediately respond to take down the bad guys before anything serious happens. A few virus cells may get inside you, but they're stopped almost immediately. Infection is unlikely
But over time the number of antibodies in the body wanes. The cops give up their stake out and just start patrolling the neighborhood. Now when the body is exposed to a pathogen the B cells and memory cells have to start producing antibodies. But by this time the infection has already begun. The antibodies are able to prevent the disease from progressing too far, but you will still become infected and may experience symptoms, though milder than someone with no active immunity.
Booster shots cause the body to reactivate the memory cells and increase the number of plasma cells, flooding the body with more antibodies, making for a faster immune response and less chance of becoming infected.
tl;dr: booster shots keep your immune system on high alert to prevent infection