The time average of a periodic quantity $Q(t)$ is, by definition,
$$\langle Q \rangle=\frac{1}{T}\int_0^TQ(t)\,dt$$
where $T$ is the period.
For Keplerian orbits, it is usually easiest to change the integration variable to the angular coordinate $\psi$ and express all quantities being integrated in terms of $\psi$.
The exercise thus consists of doing a variety of angular integrals. There is a typo in one of the results.