The quantization condition
$$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 $$
follows by looking for normalizable wavefunction solutions to the radial [TISE](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation). See e.g. Ref. 1 for details.  

References:

1. D. Griffiths, _Intro to QM,_ 1995; subsection 4.2.1.