We ice things for a few reasons:

1. The cold signal interferes with the pain signal.

2. We sometimes want to stop excessive swelling. Although swelling is very natural and is telling you to stop moving around and rest, sometimes we need to move around.

3. Cold application (especially when repeated over time, or alternated with heat) increases blood movement/drainage in the area and can reduce the amount of inflammatory mediators that stick around. These mediators can stick around for longer than Mother Nature intended, and that's why cold application after the acute injury (into the chronic phase) can also be helpful.

I always thought ice was to numb the area so you don't feel the pain.

Nassim Taleb has made the same observation.
It does appear we are treating symptoms rather than underlying cause when we ice injuries.