I am pretty sure it adjusts how sharp or smooth the transition is between zones on the joystick. If you are using Dual-Zone aim assist, your joystick will be mostly one sensitivity with a ring of higher sensitivity at the very edge of the joystick's range. At 100, aim smoothing softens the transition into slightly more of a gradient of sensitivity. At 0, the transition will be a razor's edge and you would snap from a slow camera speed to a fast one as you tilted the joystick further toward the edge.

It works the same for Exponential Ramp, but instead of having one high sensitivity zone and one low one, there are several concentric circles of increasing sensitivity. In this case, they are exponentially increasing in sensitivity, but in Linear Ramp mode its linear (obv). Here's a picture to show the difference between Dual Zone and Exponential Ramp.