Here's a very basic way to see this easily:
This is Action, search for the "Abbreviated Action", and it has the SI unit of Joule-second. This is the equation that Planck (and later, Einstein) used: $$E=nhf$$ for $n=1,2,3...$ and $f$ in frequency, in unit of 1/second).
This means that the Planck's constant has also an unit of Joule-second, therefore, you can interpret $nh$ as the Action of the system (since $n$ is dimensionless, it will maintain the unit, and it is only used to give the "correct" answer, since not every mechanical system has $h$ as the Action, $n$ should be used).
So $$nh=\int p\,dq$$ which is the Sommerfield rule for quantization. This is just an intuition of how it works.