When fruit becomes ripe and ready to eat, it ripens. Tomatoes will ripen after you pick them, becoming darker red and more juicy.
You can use ripen to talk about the maturing process of all fruits, and some vegetables — apples, peppers, and corn all need to ripen before they're at the perfect, delicious state for being eaten. Figuratively, you can also talk about people this way: "She is really starting to ripen as an actor!" Before ripen was coined around 1560, the verb was ripe, as in, "When the avocados ripe, I'll make guacamole."