The page where the proof is is on Google Books. A picture of the statement and the proof. I roughly paraphrase the statement of the result (this is Theorem 3 of Section 5.3.3, page 252-253 in the first edition, 266-267 in the second edition):
Suppose $U$ is bounded with $C^1$ boundary, and $u∈ W^{k,p}(U)$. Then there are $C^{\infty}(\overline{U})$ functions $u_n$ such that $‖u_n - u‖_{W^{k,p}} → 0$.
I'm quite sure I understand the proof but I am left a little anxious over the choice to use $r/2$ instead of $r$ when 'moving within $U$ to make space for mollification', and later $r_i/2$ when patching things up with a partition of unity. Can someone shed some light on this? Or does the proof work without using this smaller ball?
$V=B_{r/2}(x^0) ∩U$; Note that $\bigcup_{v∈ V}[ B_{ε}(v)+2ε e_1 ]$ is pushed up from $\partial U$ by about $ε$. The choice of $r/2$ stops this set from 'spilling out' of $B_r$. A (hopefully) convincing sketch: