I intuitively think about linear regression as attaching a spring between every point and your regression line (and constraining the spring to be vertical). When the line settles, that's your regression! Also gives a physical intuition about what happens to the line when you add a point. Adding a point at the very end will "tilt" the line, while adding a point towards the middle of your distribution will shift it up or down.

A while ago I think I even proved to myself that this hypothetical mechanical system is mathematically equivalent to doing a linear regression, since the system naturally tries to minimize the potential energy.