It means that sleeping early and waking up early in the morning makes a person healthy because its very constructive for the human body and mind. Hence, you will become wise also. If you have healthy and wise, then you can make the wealth.
What Does The Saying Mean? "Early To Bed And Early To Rise, Makes A Man Healthy, Wealthy, And Wise"
Going to bed early so that you can get up early presumes that you are working and saving your money. If you are out all night, besides spending money, you will not give your best to any task at hand, including but not limited to work. Sufficient rest does help ones brain development, so that is another take on it. Generally I think it applies to work and saving. 
Gay