joshmccraney, the equality comes from the fact that close to the Earth's surface the gravitational force can be regarded as being approximately constant, i.e., as not depending on distance as measured from the ground.
It is easy to see this if you expand the gravitational force in a Taylor Series around R_{E}, where R_{E} is the radius of the Earth. If you do that, you will get the following series:
F(R_{E} + r ) = \frac{GM_e m}{( R_{E} + r )^2} = \frac{GM_e m}{R_{E}^2} + \frac{-2}{1!}\frac{GM_e m}{R_{E}^3}r + \frac{6}{2!}\frac{GM_e m}{R_{E}^4}r^2 + \centerdot\centerdot\centerdot
So this is an expression for the graviational force close to the Earth's surface, being r just the distance or height you measure from the ground.
Ok, we now we have this infinite series, which looks kind of weird because we wouldn't like to calculate infinite terms to get the actual value of the force near the planet's surface. But, take a closer look at the terms of the expansion and you will see that all of them can be written as the first term multiplied by a constant times ( \frac{r}{R_{E}} )^n, with n ≥ 1 being an integer. How big is that number? We know that R_{E} ≈ 6.4×10^6 m, and say r ≈ 100m. If you plug these values into the expression you get:
( \frac{r}{R_{E}} )^n = ( \frac{100}{6.4×10^6} )^n ≈ 10^{-5n}
This means that the second term of the expansion (for which n=1) will make a correction to the fisrt term on the fifth decimal place - it's a very small correction! For the third term ( n = 2 ), for instance, the correction is even smaller since it would take place on the tenth decimal place! We conclude then that all these terms (from the second on) are too small for our purposes, since we are interested on the graviational force on the surface's neighborhood. We can then discard these terms and keep just the first one:
F(R_{E} + r ) = \frac{GM_e m}{R_{E}^2}
Now, because we don't like to write all these constants, we simply define g = \frac{GM_e}{R_{E}^2} and get the famous expression we all learn in high school:
F = mgZag
