Tactic 1: Decide on some concrete objective fact you want to introduce to the fiction. Ask one or more players what their relationship to it is.
Wasteland raiders: "So, Frankie, you live outside the main hardhold. How do you defend yourself from the wasteland raiders?"
One supply of clean water: "So, everybody else seems to think Rothschild is the only one with a source of clean water, what with his filtration machine. Do you get your water from him, or you got a secret?"
Imposing some past on a character: "When your father went off into the burn flats for the last time, and never came back, what did you do with the stuff he left behind?"
So what you want to do, is: state some sort of detail you want to be there, but state it in objective terms. This thing happened, this thing exists, this thing is a regular occurrence, these other people do these things, these other people are like this. Leave some room for the players to say how their characters have reacted to it, or normally do react to it. Let them add details.
Tactic 2: Ask for more detail. If a player says something either interesting, or fucking boring, ask for more detail. Let's take those separately.
Player says something interesting, ask for more detail because that's interesting! Flesh it out, connect some NPCs to it, get some triangles going, and look for where the PCs are weak and not in control.
Cybelle the Brainer mentions a mentor in some offhand comment.
MC: "Whoah, wait, what? You know somebody else with psycho psychic powers? Just one, or are there more out there? Who's this mentor? What happened to her?"
When a player says something really boring, it's probably because they haven't found something to be interested in yet. This is probably a good time to get them thinking about stuff. In my experience, there's this thing where the GM has to just keep throwing stuff at the players, hoping something will finally stick. You sort of have to use the same tactic here, but with questions, instead of having to think up plot hooks.
So, Wolf the Gunlugger is just like his name, a lone wolf. He says "oh, I probably just hunt for my food, I don't buy shit from Rothschild. Yeah, I hunt alone, nobody else. Y'know, whatever animals are there."
MC: "Wait, so you hunt alone, for whatever animals are out there? What do you do with the extras?"
Player: "Huh?"
MC: "The extra meat, you can't eat a whole deer at once, and there's no electricity so you don't have a fridge."
Player: "Uh, I guess I make jerky, then. Makes for less hunting, too."
MC: "You don't trade it or anything? How do you trade for stuff like shoes and bullets?"
Player: "I kill people."
MC: "I mean when you need to buy stuff from people who don't need you to kill anybody?"
Player: "Uh, yeah, I guess I trade them meat, then."
MC: "Cool, so you're an occasional meat supply. Are people happy to see you, bringing meat?"
Player: "Uh, yeah! Sure, the meat I hunt is better than their farm animals. So they're probably pretty happy to trade with me. It's high-barter meat! Very expensive!"
And finally you get something out of the player that you can work with. Now to set up some sort of meat-exchange scenario and make a PC-NPC-PC triangle.
You can combine these two tactics, as well. It's probably a good idea to add a little detail before you ask a question. Clarify the context of what you are asking for, like, "Can you give me more detail? I mean, considering Thing X that was said earlier, how do you do Thing Y? How does Thing Y work in relation to Thing Z, which I just made up now?" Plus, when you put forward details you're providing an example for the players to conform to, when introducing details.