This makes intuitive sense to me just by looking at the two extremes, assuming a persistent and consistent everlasting rainstorm.
If you're moving at infinite speed the raindrops are effectively held in place so they don't fall on you; you only get wet from the drops you walk into on your way to your destination. Since the space between you and the destination is the same regardless of your speed you'll always get at least this wet.
If you've moving at 0 speed you never reach your destination so you're in the rain for ever and get infinitely wet.
I'm not sure what the argument for moving slower to stay drier is.