You need VectorPoints
to be adjusted, also, gravity explodes for point mass so it is good to adjust VectorScale
and cut off the point mass with RegionFunction
:
VectorPlot[-#/Norm[#]^3 &[{x, y}], {x, -1, 1}, {y, -1, 1},
VectorPoints -> 20, VectorScale -> .3,
RegionFunction -> (Norm[{#, #2}] > .1 &),
ImageSize -> 500, PlotRange -> 1]

In order to reproduce your plot you need to play with VectorScale
3rd element:
VectorPlot[-#/Norm[#]^3 &[{x, y}], {x, -1, 2}, {y, -1, 1}, VectorPoints -> 30,
VectorScale -> {.1, Automatic, (#5)^(1/3) &},
RegionFunction -> (Norm[{#, #2}] > .1 &), ImageSize -> 500,
PlotRange -> {{-1, 2}, {-1, 1}}, VectorStyle -> "Pointer",
GridLines -> ({#, #} &[Join[Range[-1, 2, .1], {{0, Directive[Thick, Blue]}}]]),
Epilog -> {EdgeForm[{Thick, Blue}], Red, Disk[{0, 0}, .05]},
AspectRatio -> Automatic]
