This is not really an answer, but somewhat long comment. Selberg's trace formula is, of course, crying to be interpreted by means techniques coming from physics, however, personally I did not see some text which would satisfy me in all respects. Let me suggest some steps towards physical point of view (which are standard):
use Feynman-Kac formula: Tr (e^{tH} ) = path integral_{ over all closed paths} EXP( lenght of path)
Left hand side is obviosuly $sum_{eigenvalues} e^{t \lambda_i} - the "spectral part of the Selberg's formula (at least particalar case of it)
by the Stationary phase approximation we can expect that the principal contribution comes from the extremums of the action - in this case this would be geodesics. (This corresponds to limit from quantum mechanics to classical mechanics)
Remark: Steps 1-3 are morally true for any Laplacian on any Riemannian manifold
- By some kind of trick one must show for this particular case of Riemann surfaces with hyperbolic metric, THE ONLY contribution comes from extremums i.e. geodesics. (Such things happens in many other situations - e.g. Duistermaat Heckman formula, supersymmetric quantum mechanics etc...; however for the particular case of the Selberg formula I do not know the reference where this should be exposed in the manner which I would love, but may be I just did not search enough).