This quest is unlocked upon completing the "Diving for Medicines" quest. To start it, sail to the location marked on the map in Gibara and dock alongside the pirate ship you find there. You will learn about Blackbeard's troubles and decide to come to his rescue.
Edward and Adéwalé must find and protect Blackbeard from attacking British warships.
Sail southwest. When you get near Mariguana Island, you will see Blackbeard's ship anchored near the beach, with explosions suggesting a heavy fight is under way.
Drop anchor at the marked location on the west side of the island.
Go ashore and run to the beach near the Queen Anne's Revenge. Blackbeard and a few of his crewmen are being attacked by a group of British soldiers. Help him get rid of them. This will be a chaotic fight. You can start by eliminating the most dangerous enemies with well-aimed pistol shots. Then continue the fight in any way you choose. After some time, more British soldiers will run down to the beach from the heart of the island.
And after a little more time, an enemy galleon will arrive and a few more British soldiers will show up.
When you finally kill them all, approach Blackbeard. After a short conversation with him and Stede, the enemy galleon will start shelling you with mortars.
Board the Queen Anne's Revenge and take the wheel.
Queen Anne's Revenge has significantly more cannons than your ship and is also much more durable.
The HMS Burford is accompanied by two smaller ships. They drop quite a few barrels, so you can start by getting rid of them so they do not bother you. A well-aimed broadside volley should destroy each of them.
Right from the start of the sea battle, avoid the amber spots on the water - that is where the projectiles fired from the HMS Burford mortar will land in a moment.
Once you are left one-on-one with the HMS Burford, simply try to avoid or minimize damage from its volleys - brace, etc. - and fire at it whenever you get the chance. Take advantage of the fact that weak spots will frequently appear on its deck, and shoot them with swivel guns.
When enemy ship's health bar drops to zero, sail up to it to trigger a cutscene in which you ram it, and the boarding will begin.
You start on the mast of your ship. However, you are too far away to land an accurate shot on the captain. Jump to the beam between the mast you are standing on and the second mast, then jump to the second mast.
Next, climb down the rigging to the lower, horizontal beam of the mast, the end of which reaches near the enemy ship. From there, you can land an accurate shot on the captain.
A cutscene will trigger that will end this quest and unlock the next one - The Siege of Charles-Towne.