What was assumed to just be a wizarding fairytale in the Tales of Beedle the Bard, The Tale of the Three Brothers turns out to be based loosely on fact, and the reason that the trio learn about and eventually find all three of the Deathly Hallows.
The tale revolves around Death promising to grant the wishes of three brothers who all desire very different things, and despite each brother being given what they ask for, Death comes to eventually claim each of the brothers for himself.
A popular fan theory notices that Death, and each of the three brothers, share remarking similarities to modern day characters within the Harry Potter universe.
The oldest brother craves power, to make himself stronger than any other witch or wizard, and is granted the Elder Wand. Yet his arrogance leads to him being killed in his sleep and his wand stolen.
Lord Voldemort’s lust for immortality and power is well documented throughout the Harry Potter books. He constantly pushed the boundaries of magic, used Horcruxes to give himself immortality, and sought to find the Elder Wand himself to allow him to defeat Harry Potter – something his Yew wand was unable to do.
The second brother is deeply troubled by the death of his wife, who was the love of his life. He isn’t happy anymore, and more than anything craves to be with her, requesting the Resurrection Stone to bring her back to life. Yet despite this, she doesn’t truly exist in his world, and struck by grief he kills himself to be with her.
Severus Snape, as we discover in the Deathly Hallows, was madly in love in Lily Potter. He personally begged Lord Voldemort not to go to Godric’s Hollow on the night that she and James were killed, and became a double agent for Albus Dumbledore after that. Throughout the Harry Potter series he is always troubled and unhappy, and his death is almost a release for him.
The third brother is the only one to be wary of Death, and requests an Invisibility Cloak that allows him to hide from others and Death, far beyond the capabilities of what any spell can do. He evades Death successfully for years, and only when he is old and ready to die, does he take it off and embrace Death with open arms.
Harry Potter is thrust into situations throughout the books that he never seeks out, and despite repeated encounters with Lord Voldemort, he manages to survive all of them. He actually owns the Invisibility Cloak himself, and in the Deathly Hallows when he discovers he is the last Horcrux, he willingly allows Lord Voldemort to kill him to save the others and make Lord Voldemort mortal again.
So how does Albus Dumbledore as Death come into this?
Well, Lord Voldemort takes the Elder Wand from Dumbledore himself, yet his downfall from Harry Potter was instigated through Dumbledore teaching Harry about the Horcruxes and his immortality.
Dumbledore is also in the possession of the Resurrection Stone, and while he never gives it to Snape directly (Harry instead), Snape works for Dumbledore as a spy for the Order of the Phoenix. It is Dumbledore’s possession of the Elder Wand that leads to a path where Lord Voldemort eventually believes Snape is the true master of it, and kills him.
Dumbledore takes the Invisibility Cloak after James Potter’s death, after recognising it’s importance. He gives it to Harry Potter, and when Harry is killed by Lord Voldemort in the Forbidden Forrest, it is through his own choice after willingly following the instructions of Dumbledore who knows that Harry has to die as the last Horcrux. And who welcomes Harry like an old friend after his death but Albus Dumbledore himself inside Kings Cross station.