In the book Strauss W.A. Partial Differential Equations - an Introduction (Wiley, 2008, 1st Ed.) page $311$, there is a comment
Now an arbitrary plane domain $D$ can be approximated by unions of rectangles just as in the construction of a double integral (and as in Section $8.4$). With the help of Theorem $5$, it is possible to prove Theorem 1 (i.e. $(1)$ is Weyl's asymptotic law for eigenvalues on planar domain). The details are omitted but the proof may be found in [CH].
I'm trying actually to find the proof of $(1)$ in the book Methods of Mathematical Physics (Volume II) of Courant-Hilbert, but it is not clear where it is. Is there anyone could help me to find the page in that book?
You are looking in the wrong volume.
You should look in Vol I, Chapter VI, Section 4, Subsection 4 starting on page 436. The title of that section is "Asymptotic Distribution of Eigenvalues for an Arbitrary Domain".
Good luck!