I know what a gaussian fixed point is, and I did read the wikipedia entry, but it wasn't helpful. It says because the probability distribution is gaussian, but what probability distribution?
The probability distribution of any observable that is a linear function of fields is Gaussian i.e. $$A\exp(-(x-x_0)^2/2\Delta x^2)$$ in the ground state of a Gaussian fixed point. Also, the path integral of a Gaussian fixed point (free field theory) is the integral of the exponential of a bilinear function of the fields which is also called Gaussian.