A modern "simple philosophical" explanation is that this problem can be restated as an eigenvalue problem for a compact operator in an appropriate Hilbert space, whose eigenvalues are reciprocal to those of the Laplacian. Then two properties that you stated follow from the general properties of compact operators.

This approach is due to Hilbert. He wrote a long series of papers on the subject in
the beginning of 2-th century (see also his book with Courant, Methods of mathematical physics). Modern expositions are usually based on Hilbert's ideas.

An earlier philosophy of Poincare interprets these eigenvalues as poles of certain
meromorphic function in the plane (in modern language it is essentially the resolvent), and the poles of a meromorphic function are isolated and tend to infinity.

Two remarks should be made:

a) At the time of Poincare and Hilbert, the modern formal notion of compact Riemannian manifold did not exist.  

b) There was a very large number of problems about vibrations which were solved "explicitly" in 18th and 19th century. So Hilbert and Poincare had a lot of "empirical material" to generalize.