Anakin is the real protagonist of Star Wars.
Lucas chose three-act narration when creating Star Wars for both the backstory and the main screenplay.
Lucas wanted to start the storytelling in the middle of Anakin's story.
All of this combined -- Lucas chose to use IV to represent the beginning of the middle of the story (i.e., A New Hope), and ANH is middle of the story because the story is really about Anakin/Vader.
why did George Lucas start the numbering at 4? I know (now) he had a vague idea of what would be in 1, 2, and 3. But why not start with calling the first movie 1, and come the time to do the prequels, call them whatever?
There are three things that need to be mentioned to answer this --
1. According to George Lucas, Star Wars is specifically about the tragedy of Anakin Skywalker.
From a 2008 MTV News interview where Lucas talks about Star wars: Clone Wars and how it's separate from the epic tragedy that's portrayed in the films:
"The epic itself is basically about one man. You pass through a lot of things, but you never get to look at it. [With 'Clone Wars'], we're not burdened by the mythological underpinnings. We get to go more places," Lucas said. "The story about Anakin Skywalker and his fall into the dark side and redemption by his son, that's finished. It was started when he was 10, it ends when he died. There's no more story to tell. All that stuff is really not part of what this is."
And then, from the May 2008 edition of Total Film Magazine:
Q: Are you happy for new Star Wars tales to be told after you're gone?
LUCAS: "I’ve left pretty explicit instructions for there not to be any more features. There will definitely be no Episodes VII-IX. That’s because there isn’t any story. I mean, I never thought of anything! And now there have been novels about the events after Episode IV, which isn’t at all what I would have done with it. The Star Wars story is really the tragedy of Darth Vader. That is the story. Once Vader dies, he doesn’t come back to life, the Emperor doesn’t get cloned and Luke doesn’t get married …"
So, clearly George Lucas created Star Wars for the purpose of telling Anakin/Vader's story, and not necessarily Luke's (though, Luke obviously plays an extremely important and intimate role in Vader's story).
2. Lucas's original intent was to produce just the first three films, and he wanted to start the storytelling in the middle of the story.
When Lucas first wrote the Star Wars screenplay, his intent was to start in the middle of Anakin's story. This in mind, he first created a "whole backstory" (the events in episodes I-III), and then, he wrote the screenplay for what took place in episodes IV-VI.
From a 1993 interview with Lucas (starting at 1:41 in the video):
"It started out that I was going to write a screenplay, and in order to write the screenplay, I knew I was going to kind of start in the middle of something -- I didn't wanted to just [unclear what Lucas says] -- so I had to create a world, and in order to create that world with characters and all that stuff, I had to do this whole backstory. About where they came from, who they were, and what happened, and how they got to where they are. And so then I started this movie, and I wrote the movie, only it was way too big, and it was three acts, so I mean..
So I took the first act and I said well, I can make the movie about this, because I can think I can get this done; it was sort of within the range of what I could do. So I made a movie out of the first act, and that was such a hit I was able to do the second two acts."
And it's at this point, when transitioning into The Empire Strikes Back, that Lucas retroactively [and moving forward] applies a numbering and subtitle system to these films.
Here's another interview where Lucas talks about the prequels, and how only the first three films were apart of the original plan --
So, why did Lucas use IV instead of I for A New Hope?
3. Because Star Wars adheres to a three-act structure, and again, because the story's real protagonist is Anakin/Vader.
Since Lucas specifically wanted to start in the middle of the story (a narration technique known as In Medias Res), and, with the story's true protagonist being Anakin, Lucas needed to choose a number for A New Hope that conveyed "the middle".
Star Wars matches the second style of In Medias Res.
This being said, the storyline of the backstory and of the main screenplay both perfectly fit a three-act structure. In the previous section, Lucas is directly quoted using this format when creating the main screenplay, and, in all probability, he also used this format when creating the backstory (three-act narration is extremely common).
So, since the screenplay is three acts long, and, since the backstory is three acts long, then, the beginning of the middle would be the fourth act.
When you combine all of this together, it perfectly explains why Lucas assigned the title Episode IV to A New Hope.