The fact (!) that the Laplace-Beltrami operator on compact (connected) Riemannian manifolds has compact resolvent, and therefore has purely discrete spectrum, is well known. So far as I can tell, the most common textbook or monograph discussions of things in this vein are treatments of Hodge theory, where most of the foundational analytical details and facts seem to be taken for granted.
It seems also to be well known that the idea of a proof is to smoothly cut up the given manifold into small-enough pieces that they can be continuously imbedded into Euclidean cubes, and, then, into products of circles. On the latter, with the usual flat Laplace-Beltrami operator, the theory of Fourier series easily gives a Rellich compactness lemma... which then gets transported back to the manifold.
Not that the proof details are necessarily interesting... but... who did such a thing first? I'm having great trouble following bibliographic links back to anything...