From what I can see in the image, it looks like the selected bone (light blue) is the one moving the finger.. This is not what you want. Instead of painting the whole hand to the main hand bone (like fingers etc), try painting the fingers to the finger joints, the wrist part to the wrist, and so on. Don't know if I'm explaining this properly, but this might help; If vertices are assigned to more than one bone, they will act like in the picture. The "main handjoint" will move all child bones (in this case fingers etc), which will in turn move the vertices that are assigned to them. This means that you only need to assign vertices to the individual bone that is meant to move them.

I think this might be how the mesh is rigged in the image:

Main hand bone is moved --> so all fingers need to be moved too = to do this the finger vertices were assigned to the main hand bone, since they are meant to move when the bone moves

And it's supposed to be this process:

Main hand bone is moved --> children joints are moved too (fingers etc), because they are children --> These children bones (fingers) affect the finger mesh/vertices, the main bone doesnt affect these vertices

Hope that made a shred of sense, am terrible at explaining this sort of thing.. Let me know if I got something completely wrong :)