The "link" comes from the path integral formulation of quantum mechanics.
There's a certain dictionary that maps quantities from the canonical formulation to path integrals which closely resemble correlation functions from statistical mechanics. Specifically, suppose that $\varphi_1, \dots, \varphi_n$ are $n$ values of certain physical observables which correspond to quantities measured at times $t_1 > \dots > t_n$.
A quantum transition amplitude is given by
$$ \left< 0 \right| \hat{\varphi}_1 \dots \hat{\varphi}_n \left| 0 \right>, $$
where $\left| 0 \right>$ is the vacuum state of the quantum system, and quantities with "hats" represent quantizations of physical observables (linear operators acting on the Hilbert space).
It encodes a certain probabilistic property of quantum systems. For example, for $n = 2$, its absolute value squared encodes the probability density of a transition between two quantum states.
On the other side of the correspondence is the path integral
$$ \int Dx e^{i \hbar^{-1} S[x]} \varphi_1[x] \dots \varphi_n[x], $$
where all quantities are just numbers. The expression
$$ \rho[x] = e^{i \hbar^{-1} S[x]} $$
can be thought of as the probability density functional defined on the space of all trajectories. However, the similarity is only formal: unlike probability densities, it is complex-valued, and generally ill-defined without delicate procedures called renormalizations.
This link can be made precise for Wightman QFT and statistical mechanics with Osterwalder-Schrader axioms. However, the absolute majority of realistic QFT models are based on the gauge theory, for which there's no known axiomatization, so the link remains just a vague conjecture.
Actually, making this precise for gauge theories is related to one of the millennium prize problems.
