Here is one way to show that (nm - 1) breaks are sufficient. (It does not, however, demonstrate that (nm - 1) breaks are necessary.)
The block of chocolate has (n-1) and (m-1) 'score lines' on each side, along which it can be broken. By making (n-1) breaks we can produce n long and thin pieces, of size 1 x m. We then need to make (m-1) breaks on each of these long and thin pieces to get them down to 1 x 1 size. ie, n(m-1) more breaks.
This gives us a total of (n-1) + n(m-1) breaks.
(n-1) + n(m-1)
= n - 1 + nm - n
= -1 + nm
= nm - 1