Lots of different ways to answer, but none of them can be too intuitive since imaginary time is, well, imaginary. But here is one attempt to make the result more or less self-evident.
The basic object to calculate in quantum statistical mechanics (in thermal equilibrium, in the canonical ensemble) is the partition function (with potential insertions if you want to calculate correlation functions):
$$Z= \operatorname{Tr}(e^{-\beta H})= \sum_\psi \langle \psi(0)|e^{-\beta H}|\psi(0) \rangle$$
where $H$ is the Hamiltonian and we have a sum over any complete set of states $\psi$, written in the Schrödinger picture at some fixed time which we take to be $t=0$. In that picture the time evolution of a state is
$$|\psi(t)\rangle = e^{-i t H}|\psi(0)\rangle$$
The basic observation now is that the Boltzmann factor $e^{-\beta H}$ can be regarded as an evolution of the state $\psi$ over imaginary time period $-i \beta$. Therefore we can write:
$$Z= \sum_\psi \langle \psi(0)|\psi(-i\beta) \rangle$$
This is now the vacuum amplitude (with possible insertions) which is the sum over all states $\psi$ in some arbitrary complete basis. Except that you propagate any final states with time $ i \beta$ with respect to the initial state. In other words however you choose to calculate your vacuum amplitude (or correlation function) — a popular method is a path integral — you have to impose the condition that the initial and final states are the same up to that imaginary time shift. This is the origin of the imaginary time periodicity.
Statistical field theory is equivalent to quantum field theory if you perform a Wick rotation in time. Inverse temperature $1/T$ is identified as time.
Of course, the metrics are different. In QFT, it is Minkowski while in SFT, it is Euclidean.
Best Answer
I think it will depend the kind of statistical mechanics. For classical statistical mechanics, there is no time, so it is really hard to imagine a nice physical picture of the propagation of something. But nevertheless we still talk of loops as propagating "particles" (we give the "momenta", for instance, which is conserved, etc.). Interestingly, renormalization (a la Wilson) is easier to understand on a physical ground in statistical physics, where the coarse graining has a very nice interpretation.
On the other hand, in quantum statistical physics, the analogy is bit more direct, though time is still imaginary, so nothing really propagates. But in some sense, we still sum over all the possibilities (in a static sense, though). In this case, AB effect will give the quantization of the flux, or the Quantum Hall Effect.
Anyway, concerning the first question, keep in mind that loops, Feynman diagrams and virtual particles are artifacts of pertubation theory, and therefore have no real physical interpretation.