If you’ve seen the airplane safety demonstration enough times, you know the drill for the oxygen masks. If there is a loss in cabin pressure the masks come down. First, you put yours over your face and pull the strap over your head, and then you help anyone else who needs it. Makes sense. You’re no good to others if you’ve passed out from lack of oxygen. But what about the next part, when they tell you that “although the bag on the oxygen mask may not inflate, oxygen is flowing to the mask”? If the bag doesn’t do anything important, why is it there?
The passenger airplane masks are “continuous flow” masks, which are different from “diluter demand” masks, which are what the crew have. A diluter demand mask only releases oxygen when the user inhales. This prevents waste of oxygen and allows the instruments on the mask to determine the exact oxygen level being delivered. It functions to make sure the crew stays alert enough to get the aircraft down into more oxygenated air.
A continuous flow mask is constantly releasing oxygen whether the user is inhaling or not. Passenger masks don’t have to function as efficiently because they’re there just to make sure you stay alive. Even with some waste, there's enough oxygen to sustain everyone until the airplane reaches denser air.
The bag on the mask allows for a bit of oxygen savings. If the mask is situated well on your face, the oxygen continuously coming out will collect in the bag as you are exhaling instead of seeping out the sides of the mask. (The air-flow on any oxygen mask is one-way. Exhaled breath does not travel back into the tube, but out of valves on the mask.) However, even if there is a good seal, if you are breathing fast (which is probably what’s going to happen in a panicky situation), you’ll probably suck down what’s going through the bag before it fills. They don't want people thinking, "Hey this thing's not doing anything!" and taking off the mask or freaking out if their neighbor's bag inflates but theirs doesn't. That’s why the safety demonstration wants to reassure you that, yes, your mask is still doing its job even if the bag isn’t inflating. You’ve got enough to worry about if the oxygen masks are down. Your oxygen mask bag shouldn’t be one of them.