ELI5 Once someone contracts a contagious disease, at what point and by what mechanism is that person no longer contagious to others? Does being a "carrier" affect this?

It depends on the situation.

If someone contracts a cold, they are typically still contagious until a few days after they stop showing all symptoms. So at this point, their immune system has fought off the virus, and you give it a couple days for everything to clear away, then you are good to go.

Every virus is different. You could have a virus and never get symptoms and just be a carrier. In that case it's hard to determine when you are contagious because you probably don't even know you have the virus.

There are even cases, like with Typhoid Marry, where she was a carrier of typhoid and had to be put into a hospital for the rest of her life because her immune system never fought it off and she was a danger, as a carrier, to everyone she was around.

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As the other commenter describes, different illnesses work in different ways. Most of the time, they are in the first category //u/ntengineer describes, where your immune system fights it off. Here's a bit more detail about that process:

The immune system can be divided into two parts, the innate immune system and the adaptive immune system. The innate immune system reacts first, and includes mechanisms that are meant to kill anything, nonspecifically, such as a fever. If this is not sufficient, the germ will be detected by the adaptive immune system. This involves a molecule from the outside of the germ being detected, determined to be foreign, and the body producing antibodies (protein molecules) that can attach to it. Antibodies are produced by B cells, and at any given moment you have a bunch of B cells with randomized antibodies just waiting for something that matches them. Once a match has been found, the B cell will activate and rapidly proliferate, so you have a lot of B cells making a lot of antibodies. Antibodies sticking to a germ can inhibit it by blocking it from doing things, but mainly they tag it so other immune cells can identify and destroy it. This is usually where the disease is eliminated from the body. The antibody levels start low, rapidly increase into a peak, and then slowly level off and decrease with time once the disease has been fought off (though the length of time that each step takes varies a lot from disease to disease and person to person).

Some B cells turn into memory cells that can stick around for years in your body in case you encounter the same germ again. This is what makes you resistant or immune to reinfection, and what the point of vaccines is.