This is a different version of the old egg suck into a milk bottle demo. I got this idea from a Steve Spangler demonstration in which he uses a water balloon instead of and egg. Basically a piece of burning paper is placed into a flask and then a water balloon sits on top. The oxygen gets used up during the reaction, creating a lower pressure inside the flask than outside. The balloon gets pushed in as a result.
Thanks to Joshua Buchman for correctly pointing out that I was completely wrong in my explanation. The video now has the correct explanation. The reason the balloon gets pushed into the flask is due to the rapid cooling of the hot gases in the flask once the chemical reaction stops. The gases condense as a result, lowering the pressure inside the flask and the atmospheric pressure pushes the balloon into the flask. I had wrongly assumed what had happened based on the recollections of demos that I had seen in the past.