The Metric Tensor, A Body of Mass m and Minkowski Space 
By saying the body has mass m, we mean that the metric approaches that of Minkowski space for large r and that
$$g_{00} \sim 1-2m/r$$

This was under a section in General Relativity by Woodhouse titled The Fiedl of a Static Spherical Body
I have no idea how the sentence implies the equation (so obviously)? I cannot see why we are only considering the 00 component and the right hand side I have no idea.
 A: For a Non-rotating and Electrically neutral massive body on a spherical-symmetric static spacetime, the appropriate metric is the Schwarzschild metric:  
$$ds^2 = -(1-2m/r)dt^2+(1-2m/r)^{-1} dr^2 +r^2d\theta^2+r^2\sin^2 \theta d\phi^2$$
the $00$ component of the above metric (in the form of line-element) is the factor multiplying $dt^2$, for large "$r$" i.e. $r>>2m$, this factor is $\sim 1$, then this metric is reduced to the Minkowski metric: 
$$ds^2 = c^2dt^2-dx^2-dy^2-dz^2$$
Just set $c=1$.
However, this is an exact solution, in ~1915 (Before Schwarzschild found that solution) Einstein derived an approximate solution (first order approximation) for calculating Mercury's orbit around sun assuming that mercury has a mass negligible to the sun's mass  and moves around the sun according to geodesic equations and then used perturbational methods. The solution that einstein found is a correction to newtonian-gravity, this wikipedia article explain the concept of newtonian-limit in the context of General relativity.
you can find a full derivation in this pdf.
