We are not used to energy eigenstates in real life. This is because an energy eigenstate just picks up an unobservable phase under time evolution, and therefore doesn't "physically" change. In some sense, energy eigenstates are "motionless."
While a coherent state isn't an eigenstate, it does have some nice properties. For one, even though it doesn't have a definite energy, the uncertainty in its energy is pretty small. Furthermore, they have the minimum amount of position-momentum uncertainty as allowed by the Heisenberg uncertainty principle. This is not the case for the wave functions of the energy eigenstates, which are "smeared out" roughly in the allowed classical region. They are not spatially localized, unlike the Gaussian lumps of the coherent states which oscillate back and forth at the classical frequency.
The larger question is why would you expect the energy eigenstates to look like anything you have classical experience with? In some sense, saying a quantum state "has a definite energy $E$" is just language we use to describe the math. Useful language, but language nonetheless. It's really an assertion we place into the language when we say that eigenstates of the Hamiltionian "have" a definite energy. What does it even mean for a state to "have" an energy in quantum mechanics? We have to define the word "have an energy" for ourselves.
In the end, having a decently localized spatial extent is more "classical" than "having" a well defined energy.
However, the coherent states are still "sharply peaked" around definite energy states. Because they have a pretty well defined position and momentum, it stands to reason that they have a pretty well defined energy as well. (By "pretty well defined" I mean a small uncertainty.)