This page explains how to start and complete the Balloon Flight side quest.
How to Start the Quest
Go to Woodland Stable, which is near Mirro Shaz shrine, and talk to Shamae, the girl who is near the barrels, to start the quest.
How to Complete the Quest
You will need a couple of Octo Balloons. If you don't have any, there are Octoroks in the water behind the stable that you can kill to get Octo Balloons.
Throw two Octo Balloons onto a barrel. To do this, hold at least two of them, and look at the bottom of the screen to see which button to press to switch from "near" to "far", so that you can throw them on top of a barrel. After you make a barrel float, wait for a bit, and Shamae will react. She will give you a Star Fragment. The barrel will eventually fall, so keep an eye on where it floats to avoid letting it fall on you.