I keep repeating myself in these answers. Basically Physics does not answer ultimate "why" questions. It really answers how from certain principles and assumptions the mathematical formulation that describes the data will also predict new phenomena. How, not Why.
Why questions on physics become nested as you have seen yourself, when you found an answer to your first question :
The mass eigenstates are the free-particle solution to the wave equation
you go on to ask
"why is that"
The ultimate answer to why in physics ends up in because that's the way experiments tell us that nature is.
@Lubos has answered the basic part, that a free neutrino can be in a linear combination of the allowed mass states.
Neutrino oscillation arises from a mixture between the flavor and mass eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different superposition of the three neutrino states of definite mass. Neutrinos are created in weak decays and reactions in their flavor eigenstates[nb 1]. As a neutrino propagates through space, the quantum mechanical phases of the three mass states advance at slightly different rates due to the slight differences in the neutrino masses. This results in a changing mixture of mass states as the neutrino travels, but a different mixture of mass states corresponds to a different mixture of flavor states. So a neutrino born as, say, an electron neutrino will be some mixture of electron, mu, and tau neutrino after traveling some distance.
The way we can detect neutrinos is by their interactions, and these are signed by the flavor of the neutrino when it interacts in the detector. An electron neutrino will create an event with an electron, a muon neutrino will manifest a muon in the interaction, and a tau neutrino , a tau. Neutrino mass itself, from missing mass in other interactions of elementary particles was thought to be zero, because it was within the errors of the experiments. It was the observation of oscillations that made it necessary to postulate a mass, small, for the neutrinos.
Neutrinos are not the only ones oscillating. Oscillations were first studied with K0 short and K0 long and a mass difference was detected for the two states.An article for oscillations of neutral particls can be found here.