1980s horror doesn't get much better than Wes Craven's A Nightmare on Elm Street. Though the sequels might have gone a bit off the rails, they're all still good fun in their own right. It's a wacky franchise with a lot of really fantastical elements, and the nature of the plot makes for a no-limits approach to the kills. There's no other series quite like it, and there's certainly no villain like Freddy Krueger.
Freddy Krueger is of course the main villain in A Nightmare on Elm Street and all of its sequels. He's portrayed by Robert Englund in all of the films, until the critically panned remake in which he was played by Jackie Earle Haley. Freddy is a very iconic villain, with a lot of distinguishing and well-known traits. He wears a brown fedora with a red and green striped sweater. He's covered in burns. His signature weapon, a glove with knives on the fingers. And he's a wisecracking, funny, villain that steals the show.
Though he's seen as a bit more of a goofy, lighthearted, kind of villain, his backstory is intense. And what he does, is terrifying. Freddy is somewhat of a dream demon, and he terrorizes teens while they sleep. When he kills them in their dreams, they of course end up also being killed in real life. And with sleep being vital to human life, avoiding him is relatively impossible. But what gives him this crazy and sinister ability?
Before any answers to the question can be addressed, there is one thing that's important to establish, and that is that this franchise is quite the mess. Many different directors and writers collaborated on many different movies, and it shows. Part of where it shows is in the worldbuilding, and the lore surrounding Freddy. Different films seem to adhere to different rules, so there isn't total consistency in what he can do. This makes answering any questions in regards to why things are the way they are a little tricky, but not impossible.
Freddy Krueger's backstory is a dark one. Essentially, when he was alive he was a child murderer. After he was able to evade going to prison, the parents of his victims and other children in the town decided to get their revenge. They tracked him down and found the boiler room where he took his victims, trapped him there, and burned him alive. This is of course what gives him his burned features. This is the backstory established in the first film, and it's not made clear as to why he went from just a regular person to a dream demon. Most of the supernatural rules established here revolve around when he can terrorize the teens, which is in their sleep only at this point, and how he can be beaten, but not how he gained these abilities.
Viewers have to look to the later films, specifically to the very last film, for any real answers. In A Nightmare on Elm Street 3: Dream Warriors the story of how Freddy as a person came to be is revealed. His mother Amanda was a nun working at an asylum for the criminally deranged. After accidentally being left with the inmates alone for some time, Amanda was found to have been sexually assaulted over 100 times and found pregnant. This gives Freddy the title "son of 100 maniacs" and his DNA combined with an abusive home life made him into the human monster that he was when alive. And in Freddy's Dead: The Final Nightmare it is revealed that his powers to manipulate the dream world weren't given to him until after he was killed.
The final film in the series shows Freddy as a man being killed, and what happened afterward. He was approached by Dream Demons. In this franchise, Dream Demons are little evil beings who seek out evil people to give powers to so they can break barriers between the dream world and the real world and cause a lot of chaos. Freddy Krueger, being born of evil and going on to become one of the worst things a person can be, was a perfect target. So the Dream Demons possessed Freddy, giving him the power to terrorize children in their dreams. Eliminating them from his body was also the key to his demise in this movie.
While this explanation does technically make sense, it feels like such an afterthought. It feels like such a simple story to be the basis of an entire franchise, and it should have been introduced in a much earlier film. The movie that explains why Freddy can do what he does, is literally titled Freddy's Dead: The Final Nightmare and is the final installation of a six-film series. This will be frustrating for people who like to go into things with established backstories and rules, but it's also very on-brand for A Nightmare on Elm Street and all its mess.