What is the best available less or more modern introduction to the subject?
All approaches I am familiar with are based on some Tauberian theorem: one obtains informations about various weighted counts of eigenvalues and then one removes the weights via some Tauberian theorem.
There are three implementations of this strategy that I know of.
For the case of an elliptic operator on a smooth bounded domain see
L. Garding, On the asymptotic distribution of eigenvalues and eigenfunctions of elliptic differential operators. Math. Scand. 1, 237-255 (1953).
The journal is available at http://www.digizeitschriften.de/dms/toc/?PPN=PPN35397434X_0001
More modern literature deals with more complicated cases treated by more sophisticated methods (there are books by Shubin, Ivrii, chapters in the treatise by Hormander, papers by Birman and Solomjak etc).