You can see the shadow of heat against the paint of a car move
And why is bar soap the only soap that causes such a feeling (at least to my knowledge)?
nitrochemist: The soap removes your skin oils which provide lubrication that lets your hands feel smoother. Remove the oils and you get the friction feeling.
You can buy bar soap with lots of residual oil within that acts as a moisturizer – this is called superfatting. Basically the actual soap itself will take the oils off of your skin. That’s why superfatting is so important.
maxabean: It had to do with how soap is made, a process called saponification (fat + lye (NaOH) = fatty acid salt (soap) + glycerin). When washing with soap the soap (fatty acid salt) reacts with minerals in tap water, usually magnesium or calcium, to make soap scum. The weird feeling on your hands is literally soap scum. The other answers are wrong because you can wash your hands with liquid “soap” (which isn’t technically soap but detergents) and strip your skin of oils and still not get that feeling.