Sworn Friends are something like better Friends. True Friends.
You can Sworn a Friendship with someone if you both buy the "Promise" in the Heartstring Shop and aim for each other before using it.
After that your friend should show up in your Sworn Friend list.
You can have up to 3 Sworn Friends, with the Item "Add Sworn Friends Slot" it's possible to have 5 Sworn Friends.
As already mentioned the Heartstring Shop isn't just for the Promise it's also for the whole Sworn Friend mechanic.
Because you and your friend can select Sworn-Friends-Buff, which activates when you both are on the same map (not even in the same party)
which, naturally, works for both of you.
Indescribably Wonderful - +5% Loot Luck (Fortune)
Overcoming Trials Together - +5% Loot-chance
Bright Moonshine - +5% XP from Monsters
Peerless Combat - +1% Mastery
Love is Stronger than Gold - +1% physical/magic Power
With All your Heart - +1% crit hits
Invite - [Skill] you learn the invite-friendship skill (probably a port)
These Buffs are NOT stackable, even with multiple Sworn Friends and every Buff consumes 1 Sworn-Friend-Slot, these Slots Unlocke themself if your friendship level is high enough.
Lv. 10 is the level for the 3rd and last Buff, the level can increase with doing stuff together such as Raids, Dungeons, ...
with lv. 5 you can create and set a custom title and with lv. 10 you're able to pin your friendship symbol in front of the friendship title.