The Berry has special properties: a Pokémon eating it will have its throat cleared and voice restored. They are apparently very delicious.
In the anime
In PK23, Meloetta had lost its voice and needed a Rinka Berry to clear its throat. It enlisted Pikachu and Snivy to help it find one. After Piplup and Oshawott both failed to find the correct Berry, they got the help of a Chikorita, who led them to a Rinka Berry tree. However, it had apparently been stripped almost bare by a Snorlax, who was sleeping at its base, and had left a single Berry at the tip of the tree. Just as the group arrived, the last remaining Berry fell and bounced into a hole, which was promptly covered by Snorlax rolling in its sleep. Snivy then lured Snorlax away with a Watmel Berry, and Pikachu recovered the Rinka Berry, giving it to Meloetta. Meloetta ate it and regained its voice, and used its Relic Song to cause new leaves, flowers, and Berries to grow on the tree. Meloetta then dropped three for Pikachu, Snivy and Chikorita to eat, who enjoyed them.
Later, after Meloetta left, Meowth felt depressed that he had lost Meloetta, who he was attracted to. Depressed, he leaned against the Rinka tree, and a Berry fell on his head. When he ate it, he instantly cheered up.