Why, after absorbing a photon does an atom's electron 'fall' back to its ground state (what causes it to immediately lose its absorbed energy)?
The answer by @Davidmh gives our observations from classical physics, where we formulated the quantities "energy" , "potential" etc. We observed that this was so, an apple falls, and brilliant mathematics organized these observations into equations that can predict what will happen. So the real answer to a "why" in physics is "because we have observed that it does so". Physics then answers "how" this happens, by referring to the mathematical models constructed to describe how.
Now electrons and photons are not classical particles, they are quantum mechanical entities, working with different models of nature in a different framework. Still, there is continuity in physics, and the concept of Energy and Potential energy has been retained, and at the limit classical emerges from quantum mechanical, the underlying layer.
So the answer to "why" at the electron atom photon level is really "because that is what we have observed". BUT we have a very good mathematical model to explain the "how" this happens, which is predictive of future behaviors also. This is as follows:
A photon hitting an atom has a probability to interact with it, be absorbed, and raise the atom to a higher energy level. This probability can be calculated with the tools of quantum mechanics which also predict that once the electron is on a higher energy level, there exists a strong probability that it will fall back to the ground level emitting a photon with the appropriate energy. One can think of it with the Heisenberg Uncertainty principle , too. The higher energy level has a width, delta (E) which means that it has a lifetime delta(t) of existing, it is unstable.
Our models are very good and successful, and we tend to think that they answer "why". For example in this case we will say "because the energy level has a width and from the HUP it will have to decay", but in truth, we are just answering "how" within our model this happens.
That is a general property of our universe: things tend to the state of minimum energy and maximum entropy.
If you want, the answer is thermodynamics.
This is because usually a electron can ONLY stay in certain energy states in a given atom, this is because of the quantum mechanical forces such as strong force, and Heisenberg uncertainty principles.
Let me explain further, as electron gets closer to an atom the electrostatic & other quantum mechanical forces between the electron starts pulling it, and as electron gets closer the electron is essentially more confined and since Heisenberg's uncertainty principle governs that the electrons momentum has to become uncertain so as a result the electron will not fall in because its momentum is uncertain so it will just occupy a probable space around atom, which we call a orbital\energy level and so there are only certain stable energy levels in which a electron can occupy and so if say you hit a electron with the photon the electron might for a brief time get more momentum and escape the stable orbital\stable energy level for a brief time however as the new orbital\energy level is very unstable as the electrostatic force and other quantum mechanical effects might be pulling it back so it fall back into atom and due to conservation of energy the electron emits a photon, and as it again reaches the stable configuration where the Heisenberg uncertainty principle and quantum forces again reach a equilibrium.
This is why every atom has a specific spectrum of light it also emits because of the electron falling back to a specific energy level. This is also a reason why photoelectric works!