Do the substitution as described and use the fact that ##\mu\epsilon## is small compared with ##A##.
The last equation is for the LHS only of eq.9.39 ... where ##C_I## is given in the first equation in 9.40.
No worries - the RHS in the last equ. looks a lot like the RHS in equ.9.39 so I expect that's what threw you. I suspect that Feynman exploits the similarity on the next page.