Interpretation of Poisson Summation Formula

The formula is not about the integers in the sense that it doesn't involve their multiplicative structure; rather, it's about how the integers sit inside the reals as a discrete subgroup. You can see this in the way the formula generalizes to $\mathbb{Z}^n$ sitting inside $\mathbb{R}^n$ or more generally in how it generalizes to the Selberg trace formula. The trace formula makes it clear that Poisson summation is a representation-theoretic fact; indeed one of its special cases is Frobenius reciprocity.

I want to add something to the previous answers:

Physical picture: The Poisson summation formula gives the Heatflow on the circle $\mathbb{R} / \mathbb{Z}$ when applied to $e^{-x^2 t}$ on the circle, that is a transformation law for the heat kernel

$$ \sum\limits_{n} e^{-n^2 t} = \frac{1}{{2 \pi t}} \sum\limits_{n} e^{-n^2/4 t}.$$

Number theoretic picture: $$ \theta ( t ) = \sum\limits_{n} e^{-n^2 i t} $$ can actually be defined for $\Im t >0$ and is a modular function. Using this gives nice proof of the functional equation for the Riemann zeta function.

Representation theoretic picture: The Poisson summation formula holds more generally for local compact abelian groups. The Selberg trace formula generalizes it to non commutative groups and cocompact discrete subgroups.

(Algebraic) Geometry: The Riemann Roch theorem for function fields follows from it, which can be seen as an index theorem.

If you want analogies from electrical engineering, one possible way to understand the consequences of the Poisson summation is the Nyquist sampling theorem. It says that periodic sampling of a signal is enough to capture all the signals whose frequency is less than or equal to half the sampling rate.

in QM the POISSON summation formula is interpreted as the density of energies

$ \rho (E)= \sum_{n=0}^{\infty}\delta (x-n^{2} \pi ^{2}) $ of the Hamiltonian operator

$ H=p^{2} =-\hbar^{2} \frac{d^{2}}{dx^{2}}y(x)$ with boundary conditions $y(0)=0=y(1)$