As I mentioned, looks like the author averaged over the oscillation cycle. This is useful when the instantaneous energy is not important. The form of the cycle-average he uses, $E^* E/2$, appears as a 'trick' when dealing with complex oscillating fields. Since the energy is quadratic in the fields (and thus nonlinear) one should use:
$$
u = \frac 1 2 \langle\text{Re}(D)\cdot\text{Re}(E)\rangle
$$
for the cycle averaged energy density because only the real part is physically measurable. When both fields evolve with the time factor $e^{-i\omega t}$, the cycle average is easy to calculate:
$$
\langle\text{Re}(D)\cdot\text{Re}(E)\rangle = \frac 1 T \int_T \frac 1 4 (DE + D^* E^* + D^* E + DE^*) \, dt
$$
The terms $DE$ and $(DE)^*$ oscillate with twice the frequency while $D^* E$ and $DE^*$ don't oscillate at all, thus only the last two terms survive the cycle average. The result is
$$
\langle\text{Re}(D)\cdot\text{Re}(E)\rangle = \frac{ D^* E + DE^*}{4} = \frac 1 2 \text{Re}(D^* E)
$$
and that's where the extra $1/2$ comes from.
This stored energy is the average energy the fields carry around time $t$. If the fields are purely oscillatory, the average is the same for all times. But if there's absorption or energy pumping into the fields the cycle-average increases over time. Integrating the instantaneous power will give you the instantaneous net energy in your interval. For instance, integrating from $t = 0$ to $\infty$ may give you zero net energy because on average the energy remains in the field (the same amount is traded back and forth, so the average change is zero).
