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](//archive.org/details/grundzugeallg00hilbrich/page/n98). He wrote a long series of papers on the subject in 1904-1912.
(see also his book with Courant, Methods of mathematical physics). Modern expositions are usually based on Hilbert's ideas. The notions of Hilbert space and compact operator were essentially distilled from these works.

An earlier philosophy of [Poincaré](//hsm.stackexchange.com/questions/5706/who-came-up-with-the-link-between-the-spectrum-of-an-operator-and-the-poles-of-a) 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. Poincaré was the first to prove under general conditions the existence of an infinite sequence of eigenvalues tending to infinity.
(Sur les équations de la physique mathématique, Rend. Circ. mat. Palermo, 1894 8, 57-155.)

Three remarks should be made:

a) At the time of Poincaré and Hilbert, the modern formal notion of compact Riemannian manifold did not exist. (The notion of compact was introduced by Aleksandrov and Urysohn in 1924, and the notion of manifold by Weyl 1913, and only for dimension 2). Even in the classical book on the subject by <a href="https://en.wikipedia.org/wiki/Edward_Charles_Titchmarsh">Titchmarsh</a>, Eigenfunction expansions..., 1958, the words "manifold" and "compact" are not mentioned!

b) There was a very large number of problems about vibrations which were solved "explicitly" in 18th and 19th century. So Hilbert and Poincaré had a lot of "empirical material" to generalize. Fourier should be mentioned: his work inspired Hilbert and Poincare. He solved many concrete eigenvalue problems but had no tools to attack the general problem.

c) As a physical fact, existence of infinite discrete spectrum was first discovered (for the case of a string) by a music theorist [Marin Mersenne](//archive.org/details/bub_gal_ark_12148_bpt6k5471093v) in 
1637. This started a long story of research about these eigenvalues.