The reason for this seeming leap is that the principles of logical positivism, which is the founding philosophy of Heisenberg, Bohr, and all physicists really. This states that if a question cannot be answered even in principle by some sort of experiment, then it is not a valid question, the question is just gibberish.
Consider the following question:
- In a Hydrogen atom in its ground state, where is the electron in its orbit?
Superficially, it seems sensible, doesn't it?
But how would you formulate an experiment to determine the answer? Now it isn't so clear. Suppose you shine light on the electron to try to find out where it is, then you excite the atom, it is no longer in its ground state. Suppose you shine very low wavelength light, so as not to excite the atom. Then the light scatters off the atom as a whole, and is useless for answering the question.
If you try to use a hard X-ray to localize the electron precisely, you ionize the atom. So this question is impossible to answer by experiment, and now it doesn't look so sensible. It is a valid act of positivism to assert that this question is, in fact, meaningless. The electron does not have a position when the atom is in its ground state.
But lets say you ignore the positivism, and you suppose that the electron has a secret position which is varying in time, as Bohr often did. You might believe that the orbit is periodic, so that the Fourier transform of the orbit has integer multiples of a given frequency. The observed frequency of light emitted by a moving classical object is in integer multiples of the fundamental frequency, the inverse orbital period. So you expect that the light emitted by the atom to come in multiples of the orbital period.
But the atomic transitions have frequencies which are not integer multiples of anything. So they cannot be the description of a classical periodic trajectory. But they correspond to these periodic trajectories when the quantum number is large, when the electron is orbiting far away from the proton. This much was understood by Bohr.
But for Pauli and Heisenberg, who were more radically positivist (at first, Bohr was the most positivist of all later in life), the difficulty of finding an effective procedure led them to renounce the question entirely. They rejected the idea that the electron had a position to be determined. Heisenberg then developed a detailed mathematical theory of the transitions between different atomic levels, and this theory was able to answer questions of the form "if I shine light on the atom in the ground state, what spectral intensities come out?" But this theory could not answer the question "where is the electron", because it did not have an electron position variable which made a sharp classical orbit.
The modern perspective is just an elaboration of this position. the wavefunction describes the probability of a position measurement experiment to find a given answer, or the probability of getting the answer to an energy measurement. It does not represent the position of an electron, or any other classical quantity.
The reason people believe that there are no fundamental classical quantities evolving deterministically is because of the logical positivist position that they couldn't define them operationally. Logical positivism fell out of favor in the 1970s, for stupid reasons, so it is no longer a prominent position defended in humanistic intellectual circles. This is very sad for most physicists, who are just as positivist as ever, especially considering string theory, holography, and the positivist resolution of the information loss puzzle.