Let's not forget it can also smell mighty good...
Here is what I think.
1. Some antiseptic action. I'm not sure how practical or how needed that is today, or even yesterday. I went years just shaving and rinsing, no problem. I can imagine at some point men thought they needed it and maybe they were right.
2. Make skin feel smoother right away. Your shave will feel closer as the hairs dry and consequently shrink a little. My guess is the alcohol soaks into the hair and displaces the water, causing the hair to shrink faster than if left to dry without it. There may be some affect by other ingredients that makes the skin feel smoother, I don't know.
3. Pleasant smell. I think this was more important in the past, or expected, or routine, I don't know. Most men aren't using it as far as I can tell. I had stopped using aftershave but started again when I went back to using a double edge razor and remembered I like bay rum. So, I'm using bay rum aftershave (and the classic Old Spice) because I enjoy the fragrence. That is my reason for using aftershave, what it does for me.
4. Healing. I sort of doubt this, but I'm willing to be convinced.