Not every villain gets to be that way because they were born evil. In fact, some of the most compelling villains started off as good people - heroes even - but were warped by circumstance into doing terrible things. When viewers first meet these characters, they appear to be horrific monsters with no remorse, but once you get a peek into their backstory, your opinion of them may change - sometimes dramatically.
Obito Uchiha of Naruto may have done some atrocious things, but he also endured immense suffering. His ancestor Madara used to manipulate him into believing that humanity should be destroyed. Revy from Black Lagoon became a mercenary after an encounter with the police taught her that the only person who was going to protect her was herself. Even Cells At Work uses this trope by making viewers cry their eyes out over the backstory of a cancer cell.
Fallen hero anime characters can be some of the most fascinating characters the anime world has to offer. They offer a glimpse into what creates a villain, and how the human psyche processes trauma. Be warned - some of these stories are pretty brutal.
About half of the villains in Naruto were corrupted in some way, but perhaps one of the most dramatic examples involves Obito Uchiha.
When Obito Uchiha first appeared, he was a member of the Akatsuki who was hellbent on destroying all of humanity - basically he was as evil as it's possible to be. But Obito wasn't always like that. Actually, he was a heroic, upbeat kid who just wanted to be a great ninja. After an accident leaves his body half crushed under a rock, he gives his sharingan eye to his teammate and rival Kakashi as a final act of friendship. Kakashi and their other teammate Rin leave him, assuming that he'll be dead within minutes.
But Obito doesn't die - instead his ancient relative Madara finds him and nurses him back to health... all while slowly manipulating him into seeing humanity as a horrific thing that must be destroyed. He finally succeeds when he implants a tailed beast into Rin, which goes on a rampage that forces Kakashi to kill her in order to protect the village. Madara also arranges it so that Obito will watch Kakashi kill Rin, but not know why.
He finally gives into the months of brainwashing, and agrees to help Madara with his nefarious plans. Also - Madara's ideas about the nature of humanity were derived from a childhood dominated by war, so he could easily fit this trope too.
When Scar first appears, he's a serial killer who targets Amestrian people - Amestria being a fictional, vaguely European nation in Fullmetal Alchemist. Though a murderer when he first appears, Scar didn't always have a heart full of hate. The reason he specifically targets Amestrians is because Amestrian soldiers nearly wiped out his people, the Ishvalans. He not only lost his family, but his entire community and way of life. He wanted them to experience the same pain that they caused him - and it's kind of hard to blame him.
As Scar's arc progresses, he learns that although his anger is justified, he no longer wishes to contribute to the cycle of hate. Instead, he will work toward peace.
When Ultear Milkovich first appears in Fairy Tail, she seems like a cruel villain who is willing to kill basically anyone to get what she wants. But in fact, Ultear isn't intrinsically evil. Actually, she's acting on the assumption that she can turn back time and fix everything she's done. Why does she want to turn back time? Because of all the suffering she endured at the hands of a malicious organization that not only made her believe that her mother had abandoned her to their mercy, but who performed cruel and violent experiments on her. Had she never suffered all this turmoil and pain, she likely would have been a member of a heroic guild from the start.
Many of the villains in My Hero Academia got to be that way because they became disillusioned by hero society. Tomura Shigaraki was the grandson of one of the greatest heroes in series' history, Nana Shimura. He could easily have become a hero himself, but his life course changes dramatically when he's attacked by a villain as a young boy. Up until that moment, he had lived life assuming that if he were in serious trouble, a hero would come and save him. When this doesn't happen, he loses his faith in the hero system - and falls prey to All For One, a villain who makes him his protege. By the time he appears in the series, he's more interested in destroying civilization than protecting it.