If the force of the egg striking the floor or ground is greater than the force that the egg shell can exert on itself to keep its shape, the egg will break. Some eggs are stronger than others, depending on how thick the shell is. Certain chemicals, such as the pesticide DDT, are known to cause egg shells to be thinner and easier to break. I don't know what is "normal" for different species of birds.

The ground pushes on the egg and breaks its shell. It's like when you hit an egg against a plate or something and its shell breaks from the force and work of the hitting. Students of all ages do experiments where they try to protect an egg so it won't break when they drop it. The physics students at my university drop their eggs from the 6th floor of the physics building.

An egg, and anything else for that matter, will break ("fail") when the force applied is greater than that required to cause a loss of load-bearing capacity. Often this corresponds to the formation and growth of cracks. (Hopefully this is intuitive - once the material separates into two pieces (cracks), load cannot be transferred from one to the other.) Eggshells fail in a brittle manner (more or less), so the force to cause the formation of the crack is larger than the force to keep it growing: once a crack starts, it won't stop even if the load is removed. (There is much more happening at the atomic level, but I think it is beyond the scope of this question.) For objects of the same material, size matters - a small object can't hold up the same force as a large one. So to compare loads across size scales, force F is normalized by the area it acts on A to give a quantity called stress, σ=F/A. In these terms, a material (such as eggshell) will break when the stress exceeds the failure stress (sometimes called strength) of the material. Note - it is still force that causes failure; rewriting it as stress is a way to compare the relevant forces at different sizes.

The force required to break the egg can be estimated using a bit of physics.

If an egg breaks by dropping it from a height of 4 ft, the force applied to the egg must be greater than what the eggshell can withstand. In this case, the force is from the collision of the egg with the ground. For collisions, forces are governed by conservation of a quantity called momentum p. The momentum of an object with mass m is given by mass*velocity, m*v. In a collision, the force F on an object of mass m can be calculated by manipulating Newton's 2nd Law, F=m*a, into a slightly different form. Acceleration a is defined as the change in velocity dv that occurs over some time t. So, Newton's 2nd Law can be rewritten as
F=m*(dv/t), or F*t=m*dv.
This second form is called the Impulse-Momentum Change equation, because F*t is a quantity called impulse, and m*dv is the change in momentum (as follows from m*v being momentum). To reach a specific number for F, values for each of these terms need to be given. The mass of an egg m is around 50 g (assuming a medium-large egg in the USA), and the time of the collision t must be estimated, which I'll take as 0.01 seconds. The change in velocity dv is the difference between the velocities before and after impact. After is easy - 0 m/s. The velocity must be estimated, which can be done with more physics.

The energy of the falling egg just before it hits the ground is all kinetic energy ( simple, more detail ), given by
W = K E = 0.5*m*v²
(m = mass of egg, v = velocity). But, assuming the egg was dropped from a standstill (i.e., had no velocity at 4 ft above the ground), all of this this kinetic energy used to be entirely potential energy ( simple, more details ). The (gravitational) potential energy of the egg at 4 ft from the ground is given by
P E = m*g*h (g = acceleration due to gravity, h = distance above ground). Putting it all together and rearranging a bit produces
v = (2*g*h)^0.5. Now, using known values of g = 9.81 m/s² and h = 4 ft = 1.22 m, v = 4.89 m/s.

When you drop the egg, it accelerates due to gravity. This means that the force of gravity pulling on the egg will cause the egg to go faster. This means that the egg will gain momentum (which is a fancy word for speed) as it falls. When the egg hits the ground after falling the 4 feet, the momentum very suddenly goes to zero since the egg stops moving. This causes the egg to experience a very large force from the ground. This large force causes the delicate egg shell to break. Why does the sudden change in momentum cause a large force? Well momentum and force are related by a quantity that physicists call impulse!

Basically if something experiences a very sudden change in momentum, that corresponds to a large force. If something experiences a gradual change in momentum, the force is smaller. Think of how if you are in a car and it stops quickly you might feel like there is a big force pulling on your seat belt. On the other hand, if the car stops slowly you might not feel too much. It is the same idea!