Sometimes it doesn't matter how evil a villain is - their miserable childhoods make it impossible not to sympathize with them. Whether they endured abuse, human experimentation, slavery, loss of loved ones, displacement, or any other series of devastating events, these characters have had experiences that warped their worldview towards villainy. While a traumatic childhood doesn't excuse wrongdoing, it can explain it.
Anime villains with sad childhoods appear in series of all genres. From the shonen genre, there's Itachi Uchiha of Naruto, whose war torn childhood made him an easy target for truly villainous characters like Danzo. From the shojo genre, there's Yut Lung Lee of Banana Fish, whose family history convinces him that his best option is to side with Dino Golzine.
Which of these villains did you sympathize with?
Pretty much every villain from Naruto has a tragic backstory that explains their behavior, but Itachi's is one of the most resonant.
As a child, Itachi's life was defined by constant warfare. What's more, he was working as a chuunin by the age of ten, which meant that he was participating in dangerous battles on a regular basis. Though his skills were precocious, his emotions were much the same as any other kid - he was traumatized by the violence and would do pretty much anything to prevent it.
That's why he was so vulnerable to Danzo's manipulation. Danzo approached the then thirteen-year-old boy and gave him a choice: eliminate his family and save himself and his brother, or die alongside them in a Root invasion. Itachi realized that he could have probably warned his family and prepared to fight back, but he also realized that that would probably start a war like the one that ravaged his childhood - thus taking more lives in the process. So he took Danzo's deal.
Crona never actually wanted to hurt anyone, but they weren't given much of a choice. Their mother, Medusa Gorgon, replaced her blood with "black blood" a substance that made them significantly more powerful, but also negatively impacted their mental state. As a young child, they were forced to destroy small animals in order to prepare for violence against humans. Their own will was irrelevant, and their only value to their mother was their ability to help her pursue her dangerous goals.
Because Maka and her friends were able to recognize Crona's victimization, they helped Crona achieve some level of personal freedom. Crona ultimately became one of their allies - but it wasn't an easy process.
To say that Lucy has been through a lot is putting it mildly. She's a Diclonius, which is an evolutionary offshoot of humanity that's characterized by powerful psychokinetic abilities. As a child, she was subjected to horrifically painful experiments at the Diclonius Research Institute. When she escapes, she has to take a whole lot of lives to do so.
What's more, she also spent time in an orphanage trying to live like a normal human being. Because of her horns, she experienced severe discrimination and bullying. She was unable to tolerate it anymore when a group of kids held her down and destroyed a dog she loved in front of her. She ended up committing a massacre and awakening her Diclonius voice, which urged her to eliminate humans for her own safety. However, Lucy wouldn't always be able to differentiate between people who were an actual threat to her and people who were totally innocent or even helpful, which is how she became a villain.
Vegeta doesn't actually stay a villain - he has a redemption arc and ends up joining the good guys - but he spends a long enough time engaged in evil-doing that he definitely qualifies.
As a child, Vegeta was forced to work for Frieza under threat of his father's demise. But enduring backbreaking labor under extreme duress wasn't the worst thing that would happen to the Saiyan prince. Eventually, his father - and the rest of his species - was wiped out when Frieza decided to destroy his planet.
Vegeta knew that he wouldn't be able to defeat Frieza yet. Part of his motivation for his villainous behavior was to obtain the Dragon Balls and become strong enough to challenge him.