The last part of Tate'sThesis

Is there a comprehensive reference dealing with the last part(theory in the large!) of Tate's thesis? Why is the group of S units modulo the roots of unity a free abelian group of rank m?

An adelic proof of the Unit Theorem for $S$-integer rings can be found as Theorem 8 in these notes. (You will find an attribution to Ramakrishnan-Valenza's Fourier Analysis on Number Fields. I have found this text to be readable and useful in general, but not completely reliable on the details. For instance, if memory serves I actually had to fix a false lemma in their text in order to carry out their proof, but having done that their proof is a very nice one.)

By the way, although I trust you that it appears in there somewhere, the Unit Theorem is not the meaty part of Tate's Thesis. It's the stuff about Hecke characters, L-functions, zeta integrals, analytic continuation, etc. which was then novel and continues to be of the utmost importance today.

By the way, the last chapter of the book of Ramakrishnan and Valenza contains a working through of Tate's thesis, so you might look there to see if it helps you.