Ok, let me get a bit more technical here, hope I can explain how it works in a still easy way.
So the wretched thing is a piece of rubber that basically hangs on the tooth. Which is silly because a floppy piece hanging down from a tooth won't help anyone and won't create a barrier. So there is a squared frame with only three sides and the edges of this floppy part get fixed to that. Once this is done, it is like a barrier resting on your mouth and a part of it is attached to the tooth (a lot of 3D thinking necessary here and explaining it is already more complicated than the wretched thing itself
Now our faces are not flat (thank goodness) so if you would take something totally flat such as a notebook or a piece of cardboard and let your lips touch it, you would find out that the corners of your mouth won't touch it unless you bend it. There is the U-shape that would make the corner of our mouths point away and create a space on both sides. And that's how the rubber dam sits - it is not card, but the frame makes it flat so it will touch your lips in the middle of your mouth, but not the corners so you can still breathe with your mouth. And again, here is the biggest advantage - during treatment, you normally can't breathe through your mouth during drilling because your mouth is full of water (and suction instrument). With rubber dam, any water is behind the barrier so you can breathe and swallow the whole time, no matter how much water is involved, behind the dam.
The best idea is to have a chat with your dentist, agree on a stop signal and let them put it on for a moment and give you a moment to try it out. If you don't like it, you can ask them to take it off and think of how you would like to go about things.