To "pull something" usually means to pull it and cause it to move in the direction pulled. To "pull on something" means to exert pressure on it in that direction, but it doesn't necessarily mean that the object actually moves.
For example if someone says "I pulled the lever down," one would assume that they pulled the lever and caused it to move down. If they said "I pulled down on the lever," it could mean that they pulled the lever in a downward direction, but the lever failed to move.
Generally speaking, the "on" is usually not necessary, as pulling something can also mean exerting pressure without causing something to move, as in:
I pulled the handle but the door didn't open.
I pulled on the handle but the door didn't open.
However, when you want to indicate that the object was moved, it's best not to use "on."
I pulled the box across the room
One would interpret this as saying that the box was moved across the room by pulling.
I pulled on the box across the room.
This sentence sounds unnatural, because it could mean that the person exerted pressure on the box in different locations across the room, but it doesn't imply that the pulling caused the box to move across the room.
In your example, the announcement is instructing passengers to pull the red tabs in a downward direction in order to cause the vest to inflate, even if the red tabs don't seem to actually move down permanently.