Why does microwave heat up things so much more quickly than visible light?
Microwaves don't have more energy, they just resonate at the frequency that causes molecular bonds to rotate. This specifically applies to dielectric molecules, molecules like water that have electric dipoles.
From Wikipedia:

Water, fat, and other substances in the food absorb energy from the microwaves in a process called dielectric heating. Many molecules (such as those of water) are electric dipoles, meaning that they have a partial positive charge at one end and a partial negative charge at the other, and therefore rotate as they try to align themselves with the alternating electric field of the microwaves. Rotating molecules hit other molecules and put them into motion, thus dispersing energy. This energy, when dispersed as molecular vibration in solids and liquids (i.e., as both potential energy and kinetic energy of atoms), is heat. Sometimes, microwave heating is explained as a resonance of water molecules, but this is incorrect such resonances occur only at above 1 terahertz

For reference, energy is related to wavelength by the formula:
$$E=\frac{hc}{\lambda}$$
and
$$\lambda=\frac{c}{f}$$
where

$\lambda$ is wavelength
$c$ is the speed of light
$f$ is frequency
$h$ is Plank's constant
$E$ is energy

You can see by the first equation that as $\lambda$ gets smaller $E$ gets larger.