The crowns denote your ranch's Prestige. Prestige determines how fast new visitors will appear. You can increase your Prestige with decorations.
If you want to check your Prestige, you can do so in the main building. Just tap on it and select the button with the house, and go to the second tab (with the crown). This will show you the total amount of Prestige you have, as well as a bar with a pin that shows you how well the farm is doing with Prestige for its level - obviously, you want to be in the green area for an optimal game experience.