You feed your Chao fruit to increase it's stats, replenish it's HP, and keep it from being hungry. In the garden the only fruit that grows on the trees are Coconuts. In the VMU Chao mini game there are 9 more fruits that can be obtained.
You can obtain fruit in the VMU game Chao Adventure by winning the fruit matching game, winning fights and opening chests. If you still have fruit in your inventory when you bring the Chao back to the garden from the VMU, all the unused fruit will pop out with it!
In this original Sonic Adventure game, Chao always eat all of the fruit and do not leave any behind. Also when you exit the garden, all fruit and animals on the ground will disappear! You can't store fruit for later.
When shaking coconuts down from the tree's in the garden, take care as the falling fruit can hit your Chao and hurt it! Fruit also increase the tiredness of your Chao. If your Chao wakes up from sleeping, and you feed it 4 garden coconuts it will go straight back to sleep! Lastly, feeding the Chao coconuts will speed up it's lifespan and therefore make it evolve quicker.