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Me = 4/3 π Re³  * d,   where  Re = Radius of Earth   
            Me = Mass of EArth         d = density of Earth  (we assume it is uniform)

Let us find the gravity at a distance r from the center of Earth.  Mass of Earth enclosed inside the radius r is :

M = 4/3 π r³ d  = 4/3 π r³ (3Me /Re³ 4π)
    = Me r³/ Re³

Gravity at a location r distance away from center of Earth is = G (Me r³/Re³) / r²
  Thus  g' = G Me r / Re³ = g r /Re ,
              where g = acceleration due to gravity at the surface of Earth.
       Thus  if r = 0,  g' = 0.

In other words,  M  varies as cube of R  and  in the denominator, we have a square of R.  Thus g is proportional to R inside the Earth's surface.

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