If the waves in an MRI can go through our body, why is it that light with its magnetic fields gets stopped at our skin?

In an MRI, there are two fields being applied. One is a purely magnetic DC field, with an intensity of something like 1 T. The other is a radio-frequency pulse. The RF pulse is an electromagnetic wave, so it contains both electric and magnetic fields. The frequency of the RF pulse has to be at the correct resonance for protons in the magnetic field that's been applied. If the DC magnetic field is 1 T, then the RF pulse has to be about 40 MHz.

The DC field penetrates the human body because human flesh doesn't have strong magnetic properties such as magnetic permeability.

What about the RF pulse? The human body is mainly composed of salt water. The ions from the salt make it a relatively good conductor. When an electromagnetic wave enters matter that is a fairly good conductor, the electric fields in the wave produce oscillating electric currents. These currents then produce heating, just like any current flowing through a resistive medium. The heat energy has to come from somewhere; it comes out of the energy of the wave, and the wave is attenuated exponentially. The strength of the wave falls off exponentially with depth, the characteristic length of the exponential (called the skin depth) being $\delta\approx\sqrt{\rho/\pi\mu_o f}$, where $\rho$ is the resisitivity. Typical human body tissue has a resistivity of about 1 $\Omega\cdot\text{m}$. Plugging in numbers, we get a skin depth for an MRI pulse of about 0.1 m, which is enough to penetrate pretty deep into a human body without losing too much energy.

If the waves in an MRI can go through our body, why is it that light with its magnetic fields gets stopped at our skin?

The frequency of sunlight is about $10^{15}$ Hz. This much higher frequency gives a much smaller skin depth.