The Apodization Factor determines what the Gaussian beam’s intensity profile looks like across the aperture (or the entrance pupil of the system).  More specifically, it determines how fast a Gaussian beam’s intensity falls off within the aperture.  A simple way to think about it to ask the question:  Where do I want the 1/e^2 intensity point of my beam to be relative to the edge of the aperture? For example, do I want the wings of the Gaussian to be cut off, or do I prefer to have the wings contained well within the boundary of the pupil?  After some basic math (see below), the answer is rho_0 = 1/sqrt(G), where rho is the normalized radial pupil (aperture) coordinate, rho_0 is the specific value where the Gaussian intensity falls to 1/e^2, and G is the corresponding Apodization Factor. 

So, if you have G = 6.5, then this means the 1/e^2 intensity point of your source will reside at rho_0 = 1/sqrt(6.5) = 0.39.  This is a normalized radial coordinate value.  With an EPD = 25 mm, the aperture radius is 12.5 mm.  So, in absolute terms, the 1/e^2 intensity point of the beam within the pupil corresponds to (0.39)(12.5 mm) = 4.9 mm.  If you define the “beam size” to be the *diameter* of the Gaussian beam as measured between the 1/e^2 intensity points, then the beam size is 2(4.9mm) = 9.8 mm.  (However, I think it is best to use the terms “beam diameter” or “beam radius” when discussing Gaussian beams because “beam size” is somewhat ambiguous.)

Some more details can be found here: Gaussian Beam Apodization in Zemax