Facepalm

Rinse aid is what we call a surfactant. It disrupts the surface tension of water, which in turn lessens its ability to cling to surfaces.

You know how when you get a smooth surface of glass or plastic wet, there will be a lot of beads of water that just cling there and don't go anywhere? Unless they grow big enough to start finally running down the side? That's surface tension in action. Adding the rinse aid will reduce water's ability to bead up like this on dishes. Instead, water will be more likely to run down the surface in unbroken sheets instead of beading up.

The primary intent is that more water will simply drip off the dishes due to gravity. This does make dishes come out dryer after a drying cycle, and/or decreases the time the drying cycle takes or the energy it requires to get the same effect. But the main reason wanting water to drip off of dishes is to prevent limescale on them.

When water evaporates, only the water disappears into the air. Anything that was dissolved in that water gets left behind. If your water is hard, that will mean there's a bunch of calcites that will stay behind as a whitish powder called limescale. So if you wash dishes with hard water, let the rinse water stay beaded up on them, and dry it out via only evaporation, you get some limescale buildup on them in the form of so-called "water spots".

If instead you add rinse aid, more of the water will drip off the dishes, taking all the dissolved calcites with it. Less water has to evaporate, fewer calcites are left behind on the dishes, so less limescale and fewer water spots. Thus why many brands of the stuff show photos of crystal-clear glass on the box. A water-spotted glass will be cloudy and speckled. Rinse-aided glass will--supposedly, anyway--be clearer.