The most natural way is to study the short-time asymptotics of the heat or wave kernel on M.  For example, you can use the heat kernel $p_t(x,y) = \sum_i e^{-lambda_i t} f_i(x) \overline{f_i(y)}$ where $f_i$ are the eigenfunctions with eigenvalues $\lambda_i$.  This is a fundamental solution to the heat equation.
 
When $t$ is small then you can construct a good approximation to $p_t$ near any particular $x$ by hand, using Fourier analysis in local co-ordinates.  The end result is that that $p_t(x,x) \approx C t^{-n/2}$.  Now integrate this estimate $dx$, noting that $\int_M p_t(x,x)dx$ basically counts eigenvalues with $\lambda_i \leq 1/t$.