Schrodinger believed that the phisical interpretation of the wavefunction $\Psi$ was the vibration amplitude and $ |\Psi|^2 $ was the electric charge density. While no-one disagrees with his interpretation of $\Psi$, many physicists believed Schrodinger was wrong about about $|\Psi|^2$ and argued that $|\Psi|^2$ is a probability. Schrodinger never accepted this view, but registered his "concern an disappointment" that this "transcendental, almost psychical interpretation" had become "universally accepted dogma." Why did Schrodinger forever think that $|\Psi|^2$ was the electric charge density and not a probability?
Schrodinger never fully accepted the Copenhagen interpretation, that's about as far as I will agree with the question.
But Schrodinger most certainly did not continue to believe that $|\psi|^2$ was the electronic charge density past the end of 1926. This interpretation was shot down by Bohr and Heisenberg, and Schrodinger accepted the criticism. That this interpretation is impossible in Schrodinger's formalism was conclusively demonstrated first in one of Schrodinger's own papers! The wavefunction for two electrons is a function of 6 variables, for 3 electrons in 9 variables, and the dimension keeps going up as the electrons become more numerous. The physical charge density is only a function of 3 variables.
Schrodinger's cat was Schrodinger's way of acknowedging that the wavefunction was some sort of mulitple-possible-universes entity. This was sparked by a letter from Einstein, where Einstein pointed out that a powderkeg triggered to explode on an alpha particle emission would end up in a superposition of exploded/unexploded in Schrodinger's formalism. Einstein came to the conclusion that the wavefunction was too big to be physical, and must be a statistical description of some underlying hidden variables. Schrodinger dithered on interpretation, and at times agreed with Einstein, at times not. But he didn't like Bohr's instrumentalism.
In the classical field limit, Schrodinger's sharge-density interpretation is correct, and the classical Schrodinger field has a charge density of $|\psi|^2$. The classical field interpretation is also why there's a local conservation law for the probability density (it doesn't need to be locally conserved in quantum mechanics--- only globally conserved).
I believe Schrodinger's interpretation was somewhat like an early version of Everett, something similar to Wigner's consciousness-causes-collapse interpretation. The modern decoherence views are essentially the same, except they pretend that the density matrix solves the interpretational problem (and from a strictily mathematical view, they are probably right).
Because it rescues Maxwell's equations and gets rid of the damned Quantum Leap. The fluctuating charge densities associated with superpositions of different wave functions correspond to exactly the sources necessary to generate and absorb the electromagnetic radiation associated with all thermal phenomena including the Black Body spectrum, as well as the photo-electric effect and the Compton effect.
EDIT: Notwithstanding the downvotes my answer has received thus far, the reasons I have given are exactly the reasons Schroedinger continued to believe in the charge density interpretation, as opposed to the probability interpretation, to the end of his life. You may disagree with the correctness of Schroedinger's interpretation, but that is not the question. The question is, why did he believe what he did; and I have answered that question.
(It is not true that Schroedinger "never budged" on the interpretation. He was indeed cowed in the late 1920's by relentless pressure from the Copenhagen school, but came back to his original views by the mid-30's, as evidenced by his withering attack on the probabilistic interpretation in the form of his cat parable. And The idea that Schroedinger is the spiritual father of the multi-worlds school is sheer nonsense.)