Why can phasors be used simplify getting instantaneous power from instantaneous curent and voltage?
Multiplying current and voltage in phasors form cannot represent the instantaneous power in an AC circuit. Specifically for a resistive load the power factor is unity (\$\phi=0\$) and the instantaneous power is:

$$ p(t) = \frac{1}{2}V_mI_m(1+ \cos 2\omega t) $$



Notice that resulting wave appears shifted vertically, not having negative values. In other words, only delivered power (positive). 
In other hand, if voltage and current are \$ V_{rms}\angle \phi_v \$ and \$ I_{rms}\angle \phi_i \$, the AVERAGE POWER \$P\$ is:

$$ P = V_{rms}I_{rms}\cos (\phi_v-\phi_i) $$
or
$$ P = V_{rms}I_{rms}\cos (\phi) $$

For a resistive load:

$$ P = V_{rms}I_{rms} $$