The [[ ]] is a form of Part. (look up Part in the help system.)
In 5 above, we transpose the 2nd part of am, which is the coefficient matrix, so columns become rows. Then append a row which is the constants, then transpose it all back so columns are columns again.
Using your equations:
In[14]:= eqs = {-19 u - 19 w + 6 x - 18 y + 10 z == -2, -15 u -
17 w - 16 x - 11 y + 11 z == -15, 14 u - 8 w + 13 x - 11 y == 16,
13 u - 17 w - 10 x + 18 y + 12 z == 17,
4 u + 9 w + 20 x - 16 y - 9 z == 13};
In[15]:= (* this contains same information as augmented array *)
s = Normal[CoefficientArrays[eqs, {u, w, x, y, z}]];
In[16]:= (* an augmented matrix *)
am = Transpose[Append[Transpose[s[[2]]], -s[[1]]]]
Out[16]= {{-19, -19, 6, -18, 10, -2}, {-15, -17, -16, -11,
11, -15}, {14, -8, 13, -11, 0, 16}, {13, -17, -10, 18, 12, 17}, {4,
9, 20, -16, -9, 13}}