The biggest problem in answering to this question is that it's not as straightforward as you might think to see the physics in the equations until you see them fully developed, and even then you're doing your best to assign a meaning. As a graduate student in mechanical engineering, this was hard for me because I was used to using my physical intuition to guide me through the math forest.
For a FEM problem applied to modal analysis, you basically start with a strong form equation. For example, a free vibration of an elastic rod is given by: $$ \lambda \rho u + (Eu,x),_x = 0,\ x \in (0, L) $$ Here, $\rho$ is the density, $E$ is the modulus of elasticity, and $u$ is the displacement. Let's apply a fixed boundary condition ($u(L) = 0$) and a free boundary condition (e.g. $-Eu,_x(0)=0$). You can transform it into the equivalent weak form by using a weighting function $w$: $$ \int_0^Lw,_xEu,_xdx\ -\lambda\int_0^Lw\rho u\, dx = 0 $$ This is where the math ends up losing it's physical interpretation. From here (and skipping a lot of rigor for brevity in this answer), we would basically restrict $w$ in a subset of the <PERSON> space, apply the <PERSON> approximation by discretizing the domain which creates the finite dimensional weighting function $w^h$ and solution $u^h$, make $u^h$ the sum of a <PERSON> function $v^h$ and a given function $g^h$, and then define the functions $w^h$ and $v^h$ as a finite sum of linearly independent basis functions (i.e $v^h = \sum{A=1}^N c_A N_A$ and $w^h = \sum_{B=1}^N d_A N_A$). By applying some linear algebra theorems and rearranging a lot, you eventually get the following equation: $$ (K-\lambda^h_kM)\psi_k = 0 $$ Here, $\psi_k$ is the kth eigenvector, which represents the displacement. It's hard to get an appreciation for the math that I've glossed over. The book The Finite Element Method has the full explanation for how this is developed (Chapter 1 goes over the basics of the method, and Section 7.3 develops the frequency analysis portion).
Just to clarify, we've simply called these matrices K and M, and the form of the equation bears a resemblance to a simple modal analysis for a beam, so we simply assign that as the physical interpretation. Really, the K and M matrices are highly dependent on the mesh and how the elements are connected. 418
These large matrices are constructed from each element in the mesh.
The discretization is really important. Physically for this problem, the nodes represent points at which we're calculating the displacement, and the elements connected to that node restrict its movement under an applied force. You can also think of the mesh for this problem as having $n$ number of rods connected to each other. In that way, each rod piece only really affects those it's connected to. Each element is represented by a small matrix. The element matrices are calculated like this: $$ m_{ab}^e = \int_{\Omega^e} N_a \rho N_b\,d\Omega \ k_{ab}^e = \int_{\Omega^e} N_{a,x} E N_{b,x}\,d\Omega $$ Here, $\Omega^e$ is the domain of the element itself. When the element matrices get assembled into the global matrix, some elements in the large matrix will have terms added from several smaller matrices. This is a representation of how the topology from the mesh affects the global matrix.
The $N_a$ and $N_b$ symbols are the basis functions that span the subset of the Sobolev space we chose. These are very important because different basis functions changes the solution. We can choose to use linear or parabolic functions, or even different types of functions, like Lagrange or even B-spline basis functions.
Again, I don't know how much FEM theory the OP has, and it's been really hard to try to sum up how all this works. This may not be sufficient, but hopefully it serves to highlight some of the theory behind the FE method. 333