These arrows work much like Light Arrows in The Wind Waker, destroying most normal enemies in a single hit (although they won't drop any items or weapons). They will reduce a Guardian's max health by a third. If used on any type of boss, the arrow will deal 50 damage in addition to the attack power of the bow used.
If used while wearing the ★★ upgraded Ancient Set, it will deal a bonus +80% Damage.
One arrow is given to Link by Beedle on his Customer Appreciation Day. It's unclear if that's first time the player talks to Beedle on a particular day of the month, or after some amount of interactions with Beedle or some combination thereof.
The best place to obtain these is either opening chests around Hyrule Castle, or just buying them from the Akkala Ancient Tech Lab. The fact that they can be found in Hyrule Castle may imply that either Robbie, Jerrin or Granté snuck into the ruins and hid them there.
The Ancient Arrow is the result of a regular Arrow being combined with an Ancient dagger, which is a secret bit of Ancient Technology discovered by Robbie; this is the reason the arrow's point opens like a switchblade when nocked in a bow for shooting. The first was created solely by him, 20-30 years before Jerrin joined him from Kakariko Village.
- As shown in the E3 2014 trailer, the arrow requires Link to pull a trigger located on its hilt to release the dagger's blade.
- It's been speculated by fans that the daggers used for the arrows, due to their tremendous power and obliterating effect, may have been used for assassination purposes originally, as no other Ancient Weapon features such an effect.
Hyrule Compendium entry of the Ancient Arrow