What follows is only an answer to the philosophical/intuitive question and follows only one specific point of view (there are probably many others).

Eigenvalue problem for the Laplace operator on a Riemannian manifold $(M,g)$ is a quantization of the problem of the classical motion of a particle "freely moving", i.e. following geodesics, on $(M,g)$. More precisely, the phase space of this classical mechanics problem is the symplectic manifold $T^{*}M$ (cotangent bundle, with its standard symplectic form) and the Hamiltonian is the function on $T^{*}M$ given by $H=|p|_g^2/2$, where $p$ is linear form on cotangent fibers.

Eigenspaces of the Laplace operator with eigenvalues less than E are quantization of the part of the phase space with $H <E$. If M is compact, then $H<E$ is a subset of $T^{*}M$ of finite volume and so its quantization will produce a finite dimensional vector space (of dimension roughly the volume in units of Planck constant $\hbar$): in particular, there will be finitely many eigenvalues below $E$, all with finitely many multiplicities. When $E$ goes to infinity, the volume of $H<E$ goes to infinity and so eigenvalues go to infinity. 

In fact, this picture tells you that the number of eigenvalues less than $E$ should be of the order of the volume of the set $H<E$, which is of order $vol(M,g) E^{dim(M)/2}$, where $vol(M,g)$ is the volume of $(M,g)$, and $dim(M)$ is the dimension of $n$. This is indeed true (Weyl law).

(Maybe a trivial comment, but one never knows: it is probably helpful to think about the case of the circle, and more generally flat tori, where everything is trivial Fourier analysis, before thinking about more general situations).