It sounds like a straightforward documentary: using newly-found footage, Todd Douglas Miller tells a story you've seen and heard many times before, the first time a man walked on the moon. But the way he uses the footage leads to something less straightforward, and if you know the basics of the story, you haven't seen it told quite like this.
Miller uses no narration. As with the new footage, he has unearthed audio that was previously unavailable, and he edits everything together to give us the Apollo 11 trip as if it were new. The lack of narration means there is nothing to tell us what we're seeing, which results in a movie that feels like we're back in 1969. If you were alive then, you will recall the wonder you felt when the mission was happening. If you're younger, you can get that feeling for the first time. Either way, Apollo 11 is less about facts and figures and more about how the trip affected people at the time.
Sometimes, Miller uses multiple angles to present imagery that seems brand new. When the lunar lander returns to dock in space with the orbiting space capsule, we see what the camera in the lander sees, while also seeing what the capsule sees. If it were done in a studio, you'd be impressed at the combination of the two. Knowing that it is real-time footage means you're impressed that they managed to come together at all.
I was ready to accept a poor image ... it's 50 years old, and footage from the moon has never looked good. But Miller applies modern techniques to make everything look so much better than you remember, and it's especially beautiful compared to what you've seen before.