Player-initiated summaries benefit everyone.
Since you're presenting information to be used cooperatively, having a player do the summary has benefits for everyone involved in the process.
- It helps everyone who didn't know for obvious reasons; now they have more information.
- It helps you, the gamerunner by telling you what the player thinks is important and worth focusing on in the information you presented, helping you run a game that can better play to the things they find interesting. They'll also probably interleave their own conclusions and concerns about the information they found, which can help to inspire you going forward.
- Lastly, since you're intending this information to be used cooperatively, giving the summary benefits the player giving it. Not so much right when they're giving it, but, well. If you presented information to them that you intended to be important and memorable, but their summary doesn't mention it at all, that lets you know it didn't stick. I mean, when you gave them this information, you didn't tear open the fourth wall and stick your head through and say "this is important, remember it for later!", right? You presented it as a thing that existed in the world and hoped you communicated its importance successfully. So having a player summarize what they know gives you an opportunity to also tell them what they missed.
That last one will probably be the biggest draw, from a player perspective. Getting the gamerunner to "check their work", and such.
"Cool, what's that look like?" feat. Universal GM Spackle
Those five words: "Cool, what's that look like?" are an excellent bridge from out-of-character description to in-character action. They turn you from High Judge GM the Merciless demanding a description to a curious friend who wants to hear you talk about your character.
Just be sure to act like a curious friend, too. I mean, don't use it after every sentence, just when your actual human brain can't understand how they just, for instance, explained the unified front of secret ape police from the crystal dimension and time-traveling birds from Jupiter in plain view in a coffeehouse.
But overall, saying "cool, what's that look like?", and asking followup questions until you're satisfied is a solid way to get people talking, if not as their characters, then at least about them.