It is clear that when Voldemort attempts to kill Harry Potter in the Deathly Hallows he instead ends up killing the part of his own soul hidden within Harry Potter. Why did this happen? Was an in-universe answer ever given?
He killed the piece of his soul by using Avada Kedavra on Harry.
Dumbledore explains to Harry that the piece of the Dark Lord’s soul inside him would have been killed once he was hit by Avada Kedavra.
“So the part of his soul that was in me …’ Dumbledore nodded still more enthusiastically, urging Harry onwards, a broad smile of encouragement on his face. ‘… has it gone?’
‘Oh, yes!’ said Dumbledore. ‘Yes, he destroyed it. Your soul is whole, and completely your own, Harry.”
That piece of soul wasn’t protected like purposely created Horcruxes, nor was it tied to life by any other means - so it died once the Killing Curse hit Harry.
Harry lived because the Dark Lord used his blood.
Harry, however, survived the Killing Curse because his mother’s blood lived on in the Dark Lord and tied him to life.
“He took your blood believing it would strengthen him. He took into his body a tiny part of the enchantment your mother laid upon you when she died for you. His body keeps her sacrifice alive, and while that enchantment survives, so do you and so does Voldemort’s one last hope for himself.”
- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)
That let Harry survive - even though the piece of soul died.