To use the Practice Torches correctly, you must be very close to the big Bonfire itself. If you notice, you're getting an error each time you try to use the Practice Torch while too far away from the big Bonfire. The error is "Requires Midsummer Bonfire".
1) Get the quest. Short quest timer starts.
2) Stand next to the big Bonfire, aim your camera towards the grouping of fire-lit braziers. The braziers are arranged in a grouping of 5: a row of 2 and a row of 3.
3) Select the Practice Torch item. Your mouse will now show a small spinning circle underneath it, on the game floor. When you see a brazier suddenly get marked with a red arrow floating above it, select that brazier by moving the green circle over it.
If you did this correctly, your character will throw a torch towards the chosen brazier. The torch can be seen flying through the air. A "Braziers Hit!" buff will appear on your character, with a count showing how many braziers you successfully hit.
If you did this incorrectly because you were too far away from the big Bonfire, then your character's arm will raise up, but nothing will fly out of your character's hand.
As you can imagine, assigning a hotkey to the Practice Torches item (as suggested by the other comments here in Wowhead) will make this quest much easier.