Well, I'm probably not quite what you had in mind, and admittedly I'm a poor example, but here's what I got for you:
When grabbing someone else's wrist, the first thing I feel is the urge to suppress the maniacal laughter that naturally arises. After that, because I'm suppressing the wrist toward their center (more or less) and not just grabbing and squeezing, I feel some kind of resistance in most people with less experience. I can feel shoulder tension to a degree. When I've had a sense of what I think of as "feeling into" the structure, I can't describe it other than I had a mental image of the structure further down the line. This was years ago when I was a semi-decent student and trained regularly though. Use it or lose it.
I have no idea how I feel, although I'm very sure "tense" would be the most common description, particularly after a decade of construction work. I have to constantly remind myself to "reset" when I'm trying (it really is just trying) to use my upper body powerfully.
In my experience completely "letting go" of the area being grabbed has had mixed results. When I was a kid I vividly remember trying to make someone who grabbed me let go by fighting with the grab; I gave up on that and by relaxing the limb and pulling from my legs and hips suddenly I was moving the bigger kid instead of the other way around. I distinctly remember feeling the flesh twist around my forearm as I pulled.
I've since tried similar relaxation, but against a strong grip all that happens is my muscles feel like they're about to rip off my bones.