The basic reason is that the n-th derivative of x^n is constant and equal to the factorial of n.
In[2]:= D[x^9, {x, 9}]
Out[2]= 362880
In[3]:= 9!
Out[3]= 362880
Then the observed result is a special case of the mean value theorem for divided differences (
Wikipedia) for a polynomial function and equidistant interpolation nodes, after taking into account the relationship between finite differences and divided differences. The proof follows from the Newton-form divided difference representation of the Lagrange interpolating polynomial.