You divide the x, y, z values by the w component to get the position in 3D space which is embedded in the 4D projective homogeneous space.
So (x, y, z, w) = (1, 1, 1, 1) and (x, y, z, w) = (2, 2, 2, 2) both refer to the same point in 3D space, (1, 1, 1).
For direction vectors, w is 0, which would cause division by zero. These correspond to points at infinity in 3D space in the direction of the x, y, z components.
That's basically it in a nutshell.
4D coordinates are used because you can represent any affine transform in 3D space with a (4x4) matrix. You can't do that in 3D space because the origin of 3D space (0, 0, 0) always maps to the origin after being transformed by a matrix (so translations are impossible in 3D space using just a matrix).
That doesn't happen in homogeneous 4D space since the origin of 3D space is represented by (0, 0, 0, 1) [or (0, 0, 0, w) for w != 0].
(0, 0, 0, 0) doesn't correspond to to any point or direction in 3D space, and is not a valid value for 4D homogeneous projective space.
So 4D vectors (x, y, z, w) with w != 0 corresponds to all points in 3D space and all directions ("points at infinity") in 3D space as well.
Hope that helps.