How do people in submarines use the bathroom?

The same way anyone else does; the only difference is the design of the toilets, and the proper nautical term is the "Head", not the bathroom or toilet.

Unlike standard toilets made of porcelain, submarine toilets are made of CRES (Corrosion Resistant Steel), and don't use a gravity flush system like home toilets, or pressure-flush like commercial toilets.

On a submarine toilet bowl, it's a pretty simple setup. There are 2 valves, one for seawater to fill the bowl, and a 3-4" ball valve that's connected directly to a 3' long handle that is manually operated.

After a user has deposited whatever contents into the toilet bowl, the seawater valve is opened by hand to let water into the bowl, and then the ball valve is opened, letting the content of the bowl drain into one of the main Sanitary tanks. In port, the tank is pressurized with hi-pressure air, and forced to the pierside sewer connection; at sea, it is just blown overboard to feed the fish.

There are a couple of "hazards" with this type of setup however.

Before the Sanitary tank is emptied in port or at sea, the sailor on watch responsible will ensure that all the stall doors have warning signs on them that tank is being pressurized, and for crewmembers to NOT OPERATE the ball valve unless they want a face full of hi-pressure nastiness.

While most of the time things go as planned without any incident, there are occasions when the watchstander who is doing the tank blow operation forgets to put up the signs and someone gets nailed, or the signs ARE put up, and a crewmember just "has to go", but goes with the notion of emptying the bowl after the blow operation is completed. The problem with that is that when you're extremely tired, after doing your business, pulling that valve handle is as automatic as pushing a regular toilet lever at home, so accidents do happen occasionally. There's nothing quite like seeing someone who's been hit in the face with seawater and crap blown out of a toilet bowl at 1000 psi.

On rare occasions, the Sanitary Tank main outlet valve will get clogged; the only way to unclog it is for someone to physically enter the tank and do it manually. I won't go into the particulars, but suffice it to say that straws are drawn, and full face masks/protective gear is worn before diving into the tank. It's the same as entering a full septic tank on land.