It increases their max stats, levels up their partner skill, as well as leveling up their work skills when you get them to 4 stars.

For instance, a 1 star Chikipi will only give you one egg each time it spawns one, but a 4 star Chikipi will give you four eggs each time. I have a ranch with 4 star Beegarde, Mozzarina, Chikipi, and Vixy.