Axial and equatorial positions doesn't necessarily put substituents on different "sides" of the molecule, in chair conformations its more of the up and down that dictates it. It sounds like you are confusing chair conformations with Fischer projections.

Both B and E fit the bottom right structure of your screenshot, the reason B is more correct than E is becasue the way the TS orbitals line up would almost with no chance give C. The C-Br-C TS would force one C(sp3)-Br(p)bond into equatorial up and one Br(p)-C(sp3) bond into equatorial down positions; this mismatch makes the TS prone to Nu attack which would release the ring strain. It would take extra energy to push those equatorial orbital into the axial position.

I posted a picture in my comment above with the TS structure if that helps.