Although theA structure theorem and a sketch proof is only sketchedare presented in Exercise 3.7Exercise 3.7 of
The K-book: An introduction to algebraic K-theory (Graduate Studies in Mathematics 145) by Charles A. WeibelThe K-book: An introduction to algebraic K-theory (Graduate Studies in Mathematics 145) by Charles A. Weibel.
I found this reference very helpful (see how. Indeed, the direct product decomposition of the unit group is made explicit there).