First of all, select the whole logo and ungroup it (since everything is a single group).
Now select the inner and outer part of the objects that are filled with white. For example select the outer (gray) part of the letter O and its inner white filling. So you have 2 separate objects selected now (the outer O and the inner white filling.) Right click > Make Compound Path.
Do this for all objects that you want the white part to be transparent.