The right most dot product represents the relative velocity in the direction of the spring axis. The magnitude then is multiplied by a constant and the force is along the spring direction. The internal damping is thus computed.
I was confused on the above comment when we went over it in lecture. I think the confusing part is why we need two copies of this normalized direction. What cleared it up for me is realizing that this is a dot product of the relative velocity (b_dot - a_dot) and the normalized direction, which becomes a scalar that magnifies the actual normalized direction (the b-a/||b-a|| on the left side).
Note that this is in addition to the k_s term we saw before in the previous slides. We sum the normal spring force (adjusted for non-zero length) with the damping correction