TL;DR: The quantization condition
$$n_r ~:=~ n-\ell -1 ~\in\mathbb{N}_0 \tag{A}$$
follows by looking for normalizable wavefunction solutions $R(r)$ to the radial [TISE](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation).

In more detail: Schematically, one first solves the $R(r)$ in the regions for small & large radial coordinate $r$. After factoring out the newly found asymptotic behaviours, one obtain a function $v(r)$, where the coefficients of its power series satisfy a recursion relation. It turns out that the series must truncate in order for the solution not to alter its asymptotic behaviour.
This leads to the quantization condition (A), cf. e.g. Ref. 1.  

References:

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