What was the origin of Bohr-Sommerfeld's quantization rules? What made Bohr and Sommerfeld think momentum and angular momentum is quantized? What is the meaning of momentum quantization in harmonic oscillator? Can we imagine it (by some classical example)?
 A: It was the correspondence principle. Originally Planck introduced the constant to solve the blackbody divergence problem, it defined the fundamental scale of action.
Now Bohr proposed that electrons around the nucleus could only exist at certain stable energy levels. These had to satisfy the Bohr-Sommerfield quantization equation.
$$\oint \mathbf{p} \cdot  d\mathbf{q} = nh, \quad n \in \mathbb{N}$$
Bohr required that his equation of QM satisfies the classical mechanics too. But it had to do it with a limit of large quantum numbers, where the scale of action is Planck's constant. This is the correspondance principle.
The reason he had to do it is because classical mechanics could not explain why the electron would not orbit as a classical planet around the star and thus constantly accelerate and radiate and spiral into the nucleus.
With this, angular momentum became quantized ($L = n\hbar$), and the orbit became a QM orbital, where the electron existed around the nucleus at a certain energy level as per QM.
http://www.sfu.ca/lux/Teaching/summary.pdf
