How did they submerge the submarine?

The simplest description is that they filled the ballast tanks with water to make the boat heavier. That would get the submarine near to neutral buoyancy. In addition, they used the bow and stern planes to put a down angle on the submarine so that their forward motion took them deeper into the water. When they reached the desired depth, they took the down angle off the boat.

However, there was much more to diving the submarine than that. Many different steps had to be completed, mostly at the same time, to dive the boat quickly:

1. The officer of the deck on the bridge sounded the diving alarm twice and announced “Dive, dive” over the 1MC announcing system.
2. The chief of the watch opened the ballast tank vents. This allowed the ballast tanks, which were open on the bottom, to flood.
3. The engines were shut down
4. The engine exhaust valves were closed.
5. The air intake flapper valves in the engine rooms were closed.
6. One of the engineers or machinist mates moved to the control room to man the trim manifold.
7. Another of the engineers or machinist mates moved to the control room to man the air manifold.
8. The electricians took the generators offline.
9. The electricians switched the power source to the battery.
10. The lookouts came down to the bridge from their perches on the periscope shears and proceeded down the hatch through the conning tower to the control room.
11. The lookouts rigged out the bow planes and manned the bow and stern planes.
12. The bow and stern planes were set to full dive position as long as the propellors are not raised out of the water by the down angle on the boat.
13. The officer of the deck verified that no one was left topside.
14. The OOD then dropped down into the conning tower and pulled the bridge hatch shut behind him.
15. The quartermaster dogged (secured) the hatch shut.
16. The OOD reported to the new officer of the deck, submerged, that the bridge was clear. He then proceeded to the control room and became the diving officer.
17. When the engines were shut down, the chief of the watch shut the main induction valve hydraulically. This was the 36-inch valve aft of the conning tower that fed air primarily to the engine rooms.
18. The cook or mess cook locked the main induction valve shut once it was closed.
19. The chief of the watch monitored the “Christmas tree” looking for all the red lights (open valves or hatches) to go green (closed).
20. The chief of the watch reported the status of the Christmas tree to the diving officer. Normally, it was already a green board by the time the diving office arrived in the control room, indicating that all critical valves and hatches were closed.
21. When the boat was at or passing through periscope depth, the chief of the watch would “blow negative to the mark.” The negative tank, which was full on the surface, was forward of amidships and provides momentum down, particularly forward.
22. Once the boat was submerged and leveled off at the desired depth, the diving officer worked to get the boat to neutral buoyancy or neutral trim quickly, so the speed of the boat could be decreased if desired. Otherwise, higher speed might have been needed so the planes had more effect in keeping the boat at the desired depth. At neutral buoyancy, the planes weren’t needed to maintain depth and could be at 0 degrees, with no rise or dive.

All of this happened at once or in rapid succession. The goal in WW2, and after, was to submerge the boat to periscope depth within 30 seconds and to be able to be at 150 feet within one minute. Most of the above steps had to be completed in 15 seconds or less to allow the boat to be at periscope depth in 30 seconds.

Modern submarines take much longer to submerge. A YouTube video shows a missile submarine taking about six minutes to dive. However, modern boats normally only dive once after leaving homeport and they do so in safe waters. As a result, there is no urgency like there was during WW2 war patrols.