Why does an electron lose energy so that it falls back to a lower energy state than just stay at the higher energy state?
Actually, this question is not as bad as the comments may make you think. It's actually quite a sensible question, because when we solve the usual, Schrödinger's equation-based, QM problem of energy states of hydrogen atom (or any other atom), we get eigenstates of the Hamiltonian we inserted into the equation. These solutions should thus be stable for infinite time.

In real world, however, an atom isn't isolated. It interacts with the electromagnetic field, and this interaction results in spontaneous emission, which is the reason why an atom can't be in excited state indefinitely.