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 goes 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).

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).