Hi I just saw the file and yes its definitely a weight painting issue, but the problem is that We cant really paint weights that well because the sides of the hands dont have many vertices that we can work with.
What weight painting does is (automatic wights basically just automatically paints the weights for you) it assigns a weight between 0 to 1 (blue = 0, red = 1 in between are various shades and the color green which is around .5) when you move a bone it moves the geometrys vertices according to the weights painted (0 = no movement, or bone doesnt have an effect, .5 = half movement, bone only has half the influence, 1= fully controlled by the bone)
Sionce your model has very few vertices (as mentioned the sides of the claw dont have any vertices on the face, it only has bordering vertices)
Automatic weight assigned the best approximation which was .5 (around green) to all the faces for all the bones, hence all the bones influence the entire mesh to some degree.
Firstly you’ll need to make manual cuts in the mesh to add more geometry to the entire claw, and then manually paint in red the areas you want the bones to affect, and paint blue the areas you dont want the bones to affect
The following explains weight painting nicely.
https://www.youtube.com/watch?v=pMIwthFmr-Q
The following is a great tip on weight painting with bones
https://www.youtube.com/watch?v=l9S-jhBIEiU
