Page 109 says:
"The players' job is to say what their characters say and undertake to do, first and exclusively; to say what their characters think, feel and remember, also exclusively; and to answer your questions about their characters' lives and surroundings."
Page 127 says:
"Ask about the landscape, the sky the people and their broken lives too ... Turn a player's question over to the group ... But especially, anything you want to know, ask. Anything you think might be interesting later, ask ... Start to cultivate an apocalyptic aesthetic in your players too."
So taken together, that's:
1. You the MC may NOT tell the players what their characters remember. You can state certain objective facts about the world and the past, but not the PCs' relationships to them.
NO: "The Mole Men come raiding, like they usually do when the weather gets hot. You guys always fight them back but they always get one or two of your people."
YES: "When the Mole Men come raiding, during the hot weather, what do you normally do?"
YES: "When the Mole Men steal one of your people, what do you normally do?"
2. Anything you don't want to make up about a character's life or surroundings, you ask the player.
NO: "Hey Orlando, are the Mole Men happy or mad to see Keeler?"
YES: "Hey Keeler, what does this Mole Man in front of you look like?"
Even this one is legit by the letter of the rules: "Hey Keeler, you see a dead Mole Man in the corner. What killed it? Can you tell?"
Conclusion: It's the MC's responsibility to PLAY the world, but not to CREATE it. That's flexible, but based on MC prompts. The MC still MCs.