I don't get it.

Same here. They could have left out the operating/neck scar stuff, and it would have been better. So all those wires and cords on her when she was in bed after returning to her 'mom and dad' all had to do with her forgetting again? Just got too weird - those parts of the plot.

The first thing you ask is never explained. We are told (through Grace's reminiscences) that everyone died at once on a Thursday afternoon but not what killed them. Personally, I don't think it makes a difference to the story.
We get a clue as to "what was going on in the town where they were operating on people" when the visitor who identifies himself as Grace's father invites Del to his town, telling him they have a team of doctors, the best in the world, "coming up with new ways to harness the complexities of the human mind." When Del arrives in Palm Springs, the same guy tells him that they purge people of unhappy memories and the negative emotions that come with them. In light of Grace's response, it's safe to say the procedure isn't altogether effective, but at least we get a sense of what they were *trying* to do.

Sorry I gave so much time to this boring movie.

As Del says, “Doesn’t matter.” Not even The Walking Dead or Night of the Living Dead ever explain. What matters is the remaining characters living in a new world of eternal present — especially in this flick.
I call this genre “People In a Box,” — the box here being more figurative.