The starting point of the exercise is the definition of the expectation value of n using the Boltzmann distribution (or the statistical physics). I also explained how that expectation value works in Interpretation of the partition function.
About the sum itself, there are a few things to notice here.
The sum is indeed of a form \sum_{n=0}^\infty n r^n, where r=\exp(-\beta \hbar \omega).
Getting n in front of r^n we can get by computing r d r^n / d r = n r^n (this part is really a very similar idea to the role derivatives play in Interpretation of the partition function).
Using the derivative identity we get
Then we carry out the summation using the expression for the geometric series.
Overall the first two exercises from the Einstein model are about the connection between what we do in the course and what you learned in the statistical physics. These are largely, if not completely, a repetition of what you have learned in statistical physics.