What is the origin of buoyant force exerted by fluids?

Archimedes deduced that "Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object". This buoyant force is physically due to the increase in pressure with depth in the liquid; for an increase in depth the pressure increases by liquid density x acceleration of gravity x increase in depth. The increase in pressure with depth is due to the weight of the overlying liquid. Using this relationship for pressure change you can prove Archimedes principle.

To really understand this qualitatively you should first convince yourself that the pressure in a liquid only depends on the height and density of the liquid above it. It does not matter which shape the container has. Therefore the total mass above it and volume do not matter, only their ratio does. Pascal's law is also important to understand.

Now consider a room full of water; the liquid would, were it not contained, have collapsed to the ground, however the walls are exerting a pressure (through normal force) on the liquid to keep it contained. This pressure that the walls exert is different at each height of the wall, because of the height of the fluid above it. The horizontal layers of the fluid also exert this same height dependent amount of pressure that the walls exert on the layer of liquid above it and below it. When you place an object in the liquid, the object thus experiences these same pressures at each point of its surface, and since the force exerted at the top is lower than at the bottom (because the height of liquid above it is lower at the top) it experiences an upward force.