Yes air is less dense than water but how does the bottle know to rise or indeed to move? It's not electromagnetism I think. Does this have a relation to gravitational forces? Is it to do with the number of protons in the molecules of the different substances?
The microscopic explanation for the macroscopic phenomenon of pressure is that momentum is transferred by electromagnetic forces from individual fluid particles (in this case, water molecules) to anything they approach closely enough, including other water molecules or the molecules of plastic that make up the bottle. The momentum transferred by a single water molecule to the bottle during an interaction depends on the details of that interaction, but you don't have to resolve those details to say something about the total amount of force--for that you only need a statistical description of the sum total of all the interactions, which is what pressure is.
So, why is the force exerted on the bottom of the bottle by all these collisions greater than that exerted on the top? Because of gravity and the mass of the overlying water, the water molecules further from the surface are packed slightly more closely together on average than the ones closer to the surface. Their mutual repulsion is therefore greater on average--this repulsive electromagnetic force grows very quickly as water molecules move closer together. The water molecules closest to the bottle are squeezed a bit closer to it where the pressure is greater (deeper below the surface), transmitting the higher average intermolecular repulsive forces there to the surface of the bottle.
Archimedes figured this one out around 250 BC. If you push an empty bottle underwater you have to raise the surface of the water by the same volume. This is the counterforce that you feel.