What is the meaning of the proverb The horse that arrives early gets good drinking water?

It has a similar meaning to 'the early bird gets the worm'. In other Words, being the first to do something gives you the advantage. The implication is that the early riser has a leg up over the lazy sleeper: more to choose from, or a better opportunity.

In the case of the horse, the first to arrive at the watering hole in the morning gets fresh clean water. But after he stomps around (muddying the water) and perhaps relieves himself (contaminating it), the rest of the horses have to take what's been left.