How does the cabin of the submarine get pressurized?

"Pressure in the Boat" is the proper term, and it is closely monitored while underway submerged. A drop in pressure indicates a leak somewhere onboard.

Once all hatches are closed and dogged, overall internal pressure is increased slightly to determine if there is a good seal on all hatches prior to diving. Air is bled into the pressure hull as necessary to do this, though the pressure isn't that much. The pressure is monitored constantly.

In emergency flooding cases, a compartment can be pressurized with compressed air to halt incoming water (or force it out).

Over time, pressure will increase as air is bled into the boat, and when surfacing and opening a hatch for the first time, crew must be extremely careful as the pressure is strong enough to eject a person out of the hatchway. All hatches have dog latches that prevent the hatch from being just being opened, and if there's too much pressure, it'll prevent the hatch from being blown open. However, it won't prevent someone from being blown into the hatch if they're not careful.