How do submarines shoot in water?

It depends on the weapon (torpedo, missile, mine, or vertical launched missile) but the 2 primary methods are water impulse ejection (torpedoes, torpedo tube launched missiles, mines) and air ejection (vertical launched missiles). Torpedoes also have the ability to swim out (engine startup in the tube) from the torpedo tube if the impulse ejection system is down.

The water impulse system is pretty simple; the tube at the breach end is connected via another vertical tube to a water reservoir that has a piston ram. When high pressure air is dumped from a flask directly into the air side of the piston, it instantly moves the piston ram, forcing the water on the other side of it out of the open tube, ejecting the weapon along with it.

For vertical launched weapons, it's simply a high pressure air bubble that ejects the missile; the rocket motor doesn't actually fire until the weapon clears the ocean surface.