There are two stories being told at the same time. One is the story of a 1950s stage production called "Asteroid City" and the other is an actual presentation of the stage play "Asteroid City".
It flips back and forth between the behind the scenes stage production and the actual play itself.
It's stilted and awkward because that was the style of many plays in that era. If you look at playwrites like Harold Pinter or Samuel Beckett, you'll get the same vibe.
Here are films of two of Beckett's better known productions, this is the sort of vibe Anderson was going for. It was called "theater of the absurd":
No, it doesn't make much sense. Why is that woman buried in dirt up to her waist in the first act and up to her neck in the second? You aren't supposed to ask those kinds of questions with these productions.
That's going to bother a lot of people.