Arrows are extremely useful against large swarms of low health troops such as Minion Horde. They cost 1 more Elixir than the Zap, but they have a larger area of effect, and also deal more damage, being able to kill Minions up to two levels higher than themselves.
The spell can be used to finish off buildings that are extremely low on hitpoints.
It is possible to go for a push using a Giant and a Prince or any viable combination, knowing that the opponent will use low-hitpoint units such as Spear Goblins, Archers, and Minions. When the opponent tries to defend the Arena Tower, cast Arrows on the enemy units and secure a push.
Keep in mind that Arrows has a somewhat long travel time and will take a bit of time to reach their target.
Arrows are a neutral counter to Goblin Barrel. When the opponent throws the Goblin Barrel, fire arrows immediately once the Goblins spawn, and the Goblins will be destroyed. Arrows has a long cast time, so firing the Arrows slightly before the Goblin Barrel's impact will make sure that no damage is dealt to the Crown Tower.
You can bait the opponent to use their Arrows using low hitpoint troops. Then, use troops which the opponent cannot counter easily without Arrows like the Goblin Barrel and Minion Horde. However, keep in mind that the opponent may also have other area damage spells or troops, such as the Zap, Fireball, and Wizard.
Arrows are a great counter card towards decks that have a majority of low hitpoint units, such as the Skeleton Army.
The Log can be an alternative to it, as they have similar areas of effect and damage. The Log costs 1 less Elixir, has pushback, and has a linear attack, but it can't damage air troops.
Arrows can one-shot a Princess 7 levels lower, Minions up to 2 levels higher, Skeleton Army at all levels, Goblins up to 3 levels higher, Spear Goblins up to 8 levels higher, Archers one level lower, Fire Spirits up to 10 levels higher, and Ice Spirits up to 2 levels higher.
The Arrows card was released with Clash Royale's soft launch on 4/1/16.