The Gaussian assumption is on the errors of the process, not the regressors. In the simplest case the model is $$y_i=\alpha+\beta x_i+\varepsilon_i$$, where $\varepsilon_i\sim\mathcal N(0,\sigma^2)$, i.e. errors are Gaussian.

The errors are not observed, so we can estimate them as residuals $\hat \varepsilon_i=r_i=y_i-(a+b x_i)$. Then we assume these are from Gaussian $r\sim\mathcal N(0,\hat\sigma^2)$

Then we write the MLE or least squares equations and get the estimates $a,b,\hat\sigma$.