Did you know who was banging on your door? People who don’t like to answer their doors, and I am one of them, almost always still look out a window or something to see who it is, just in case it’s important and not just someone soliciting.
That’s why the neighbors are upset at you. They think you saw it was them and didn’t answer anyway, because you didn’t want to be bothered and you assumed they were knocking (banging, you called it, well it was an emergency) just for no good reason.
That’s the problem. Their son was suddenly injured in an accident and the mother panicked and knocked (banging, you said, with tone of contempt) on your door for help. You didn’t answer then later said, oh I heard the banging but didn’t know what ‘it’ was, leaving open the implication that you might have seen who it was and didn’t want to be bothered.
How would you feel if the situation were reversed?
If you want to be on bad terms forever with the neighbors, go ahead.
But if the neighbors have been good neighbors these six years, then I would personally go over with flowers and explain that you didn’t know it was her, that you were busy with the baby or something (tell the truth though), and assumed it was another stranger as is common. Emphasize that if you had known it was her, you would have instantly answered and tell her you are very sorry you didn’t answer the door as her son was injured and needed help.
Sometimes you need to apologize. I am one of those who will hide in the back of the house to avoid answering the door, but if a neighbor comes and knocks, usually there’s a reason and I wouldn’t ignore them. It’s not like your neighbor does this all the time. Remember, they think you knew it was them or that you flippantly dismissed a panicked knocking on the door, oh I didn’t know what it was. I don’t blame them for being upset.