Inactivated (non-live) Vaccines work by taking a small blueprint protein of the enemy virus or pathogen (a small piece of the puzzle) and introduces this puzzle piece to our body (usually the muscle cells in your arm, thigh or bottom). This puzzle piece is foreign to the body, so it activates your cells to "present" it to your body's military/army (your immune system), who studies this puzzle piece and begins training exercises and creating weaponry (like neutralizing antibodies) to defend against this foreign invader. Because the puzzle piece blueprint is just a blueprint - it cannot infect or destroy cells - so it simply gets gobbled up by your immune system or degrades on its own in a few days.
While your immune system goes through its military training exercises, you will feel like you're getting a major internal workout, especially if it's a particularly effective vaccine. You get soreness or swelling in the arm and lymph nodes, some chills, fatigue, sometimes a fever (which is activated by your immune system) and body aches. Usually a very limited and short training exercise, most people feel better after about 3-4 days of this workout and your immune system is now newly trained, and therefore, stronger, to defend against the real, full strength, foreign invader.
Extra vaccine doses and boosters are simply allowing the immune system to re-train (continuing education, if you'd like to call it) to build more weaponry (antibodies) against the disease (as the weapons may need polishing or replenished from time-to-time).
COVID19 Vaccines are NON-LIVE, inactivated viruses. They cannot "infect" the immune compromised. This is why those who have weak immune systems or are pregnant can safely take these vaccines.
Vaccines are a major landmark of our health and wellness in modern society - it has given us an incredible tool to fight infectious disease and has increased life-expectancy and quality of life of humans over the past century.