Note that $e^{-\pi x^2}$ is its own Fourier Transform, then $$
\begin{align}
\int\left(\int f(x)\;e^{-2\pi ix\xi}\;\mathrm{d}x\right)e^{-\pi\xi^2\epsilon}e^{-2\pi iy\xi}\;\mathrm{d}\xi
&=\int f(x)\left(\int \;e^{-2\pi ix\xi}\;e^{-\pi\xi^2\epsilon}e^{-2\pi iy\xi}\;\mathrm{d}\xi\right)\mathrm{d}x\\
&=\int f(x)\left(\int\;e^{-\pi\xi^2\epsilon}e^{-2\pi i(x+y)\xi}\;\mathrm{d}\xi\right)\mathrm{d}x\\
&=\int f(x)\;\frac{1}{\sqrt{\epsilon}}e^{-\pi(x+y)^2/\epsilon}\;\mathrm{d}x
\end{align}
$$