For a submerged object, buoyant force ($F_b$) is defined as:
$$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$
Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to http://faculty.wwu.edu/vawter/PhysicsNet/Topics/Pressure/BouyantForce.html
and found the following explanation, but it seems like non-sequitur to me:
Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object.
Answer
The argument sounds perfectly reasonable.
Consider arbitrary parcel of fluid in equilibrium. It exerts downward force equal to it's weight on the surrounding fluid, and it does not move. Therefore according to the second law of motion, the downward force must be balanced by upward force of equal magnitude, the buoyant force (otherwise it would start to move, contradicting the equilibrium assumption).
The buoyant force is exerted by the fluid surrounding the parcel. Therefore if we replace the parcel with something else, there is no reason for that force to change.
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