The intuition comes from Poincare-invariant scalar field theories, where a (Minkowski-signature) Green's function is the vacuum expectation value of a product of field operators:
$G(x_1,...,x_n) = \langle vac | \phi(x_1)...\phi(x_n)| vac\rangle.$
Here, I'm writing $\phi(x_i)$ for the observable which measures the value of a scalar field $\phi$ at a point $x_i \in \mathbb{R}^d$. (This is a mild abuse of notation. If we were being careful I would have to explain that fields are operator-valued distributions, and that the Green's function is a Schwarz kernel for the distribution in $n$ variables given by $f_1,...f_n\mapsto \langle vac | \phi(f_1)...\phi(f_n)|vac\rangle$.)
In such theories, the group of translations of $\mathbb{R}^d$ has a unitary representation $U$ on the Hilbert space where $|vac\rangle$ lives, and it acts by conjugation on operators on this Hilbert space. In particular, translation by $y \in \mathbb{R}^d$ sends $\phi(x_i)$ to $U^*(y)\phi(x_i)U(y) = \phi(x_i + y)$. These translation operators also leave the vacuum vector invariant. This is where the translation formula you asked about comes from.
$\langle vac | \phi(x_1)...\phi(x_n)| vac\rangle
\\= \langle vac | U^*(y)\phi(x_1)...\phi(x_n) U(y)| vac\rangle \\= \langle vac | U^*(y)\phi(x_1)U(y)U^*(y)...U(y)U^*(y)\phi(x_n)U(y)| vac\rangle \\= \langle vac | \phi(x_1+y)...\phi(x_n+y)| vac\rangle$.
In the first equality, I used translation invariance of the vacuum: $U(y)|vac\rangle = |vac\rangle$. In the second, I inserted $1 = U(y)U^*(y)$ between every pair of adjacent field operators.
Your source may be talking about Euclidean signature Green's functions. These are obtained by analytically continuing in the variables $x_i$.
Generally, Streater & Wightman's book is a nice source for this stuff. Also, David Kazhdan's contribution to the IAS QFT & Strings for Mathematicians book.