The experimental probability distribution , for reactor neutrinos for example, show variations not explainable at first hand. Electron neutrinos appear and disappear and appear again.
Reactor neutrino experiments are always disappearance experiments, because the energies of reactor neutrinos – typically a few MeV – are not sufficient to create muons or taus, and therefore νμ and ντ cannot be seen in charged-current reactions.
>Figure 18: oscillation in the KamLAND reactor neutrino experiment. The plot shows how the survival probability varies as a function of L/E. The points are the data (with known background subtracted); the lines are the prediction from oscillation models. From A. Gando et al., Phys. Rev. D83 (2011) 052002.
What does the mixing angle, θ, represent physically?
An oscillation hypothesis can fit the data, but within the confines of the standard model the fit has to include measured quantities of the beam production. In your formula the second sine function is connected with with the experiment itself. The parameter $θ$ put by hand is the variable used that fits the data. That is the physical interpretation, it is necessary to fit the data , and it is sinusoidal because a constant will not do it, and when seeing a wave type behavior sines and cosines are used in a hypothesis.
Using the standard model with a small extension of the hypothesis that the neutrinos can oscillate between a mass eigenstate and weak interactions eigenstate allows for a small extension to the standard model by introducing a mass for neutrinos as an explanation of the physical observation.
So theta represents physically the necessary angle to fit the data, interpreted as between the mass eigenstates and the weak eigenstates of the neutrino.
Oscillations were observed early in particle physics and solved the experimental tau theta puzzle in a similar way.