The reason is that the matrix of the exterior derivative is equivalent to the transpose of the matrix of the boundary operator. That fact has been known for some time, but its practical consequences have only been understood recently.
[S]uppose you know the boundary of each
k-cell in a cell complex in terms of
(k−1)-cells, i.e., the boundary operator. Then you also know the exterior derivative of all discrete differential forms (i.e., cochains). So, you know calculus. Smooth or discrete.