It's a question where it helps to be good at visualising 3D objects which is either something you're good at or you're not. I'll try to explain it though. If you're working in two dimensions, then every vertex of a square in the grid will be 'connected' to 4 other squares, so if we were working in 2D then it would be 1/4 of an atom. However, lattices are 3D, not 2D. If you have a 3D 'grid', then every vertex is 'connected' to 8 cubes. This diagram shows this reasonably well:
On the diagram in the far right, you can see that the yellow sphere (which is at the corner of the cubes) is surrounded by 8 other cubes. Since an equal part of the atom is in each cube, the proportion of the atom in each cube is an eighth. Does that help?