The way I read this statement is the following: you can take projective 3-space, denote any plane as “the plane at infinity”, and the remainder (i.e. projective space minus that one plane) will behave as an affine space. Any projective transformation which preserves the plane at infinity will then correspond to an affine transformation.
All of the above can be readily seen when you consider coordinates. Using a simple transformation of your coordinate system, you can take the plane at infinite (your codimension-one subspace) to be $w=0$. Then any point $(x,y,z,w)^T$ in the complement will have $w\neq0$. So you can dehomogenize to $(x/w, y/w, z/w)^T$ and use that as affine coordinates.
what does 'geodesic preserving manner' mean?
The fact that they refer to geodesics instead of collinearity seems a bit strange to me, but I guess the message is the same: (projective) lines will map to (affine) lines. Which again is straight-forward given the dehomogenization interpretation above.
For the converse case, this “natural completion” refers to the union of the original affine space with elements at infinity. Expressed in coordinates, this is the step where you introduce homogenous coordinates, some of which may have $w=0$. Lines still remain lines.