The question is: how GR explains that we (and all objects) are under an acceleration of $g$ at the surface of the earth?
The argument of the link is that for small velocities compared to the light speeds and $r_s \lll r$, the time dilation is basically a function of $r$. But it doesn't answer the question above. It only notes that now there is a time dilation. As there is not such a thing in Minkowski spacetime for objects at rest in its frame, it concludes that time dilation is the "cause" of gravity.
In order to get an answer, we can write the geodesic equation for the coordinate $r$. After eliminating the null terms, using the Schwarzshild equation:
$$\frac{\partial^2 r}{\partial \tau^2} + \Gamma^1_{00} \frac{\partial t}{\partial \tau}\frac{\partial t}{\partial \tau} + \Gamma^1_{11} \frac{\partial r}{\partial \tau}\frac{\partial r}{\partial \tau} + \Gamma^1_{22} \frac{\partial \theta}{\partial \tau}\frac{\partial \theta}{\partial \tau} + \Gamma^1_{33} \frac{\partial \phi}{\partial \tau}\frac{\partial \phi}{\partial \tau} = 0$$
Now we use the argument of small velocities: $dr, d\theta, d\phi \lll dt$, so only 2 terms matter:
$$\frac{\partial^2 r}{\partial \tau^2} + \Gamma^1_{00} \frac{\partial t}{\partial \tau}\frac{\partial t}{\partial \tau} \approx 0$$
Using Schwarzshild metric:
$$\Gamma^1_{00} = \frac{2GM}{r^2}\frac{1-\frac{2GM}{r}}{2}$$
From the Schwarzshild equation, where $dr, d\theta, d\phi \lll dt$:
$$d\tau^2 \approx \left(1-\frac{2GM}{r}\right)dt^2$$
Leading to the desired $g$ acceleration:
$$\frac{\partial^2 r}{\partial \tau^2} \approx -\frac{GM}{r^2}$$
