Iirc through the context and how Luthien reacts. Why would this elf princess run after this particular human, and when she reacts, he can read that. Sauron still for quite a while is in his manipulator era, so i'd expect him to read ppl very well. And as for the quest, idk that's an interesting question, what exactly he thinks they're here for. But obv. an elflord /w entourage doesn't just walk into Angband for fun.
I think in Finrod and Beren's case he was just listening a lot and when they, especially Finrod in the battle, were making a move, he used that information and energy in his favour (like in judo). Maybe from listening to them he also has some kind of intuition about what they're here for. With Luthien he tries the same in the beginning, and has some amount of success with it, like with finding out Beren is a lever for Luthien, but then Luthien starts listening to him, he goes on the offensive and the same happens to him, that he volunteers pieces of info that she can use against him. I think that was discussed on the pod too, and back when the pot of evil was first discussed, i remember thinking that Luthien must somehow find that out during the song duel, and that will be her alqualonde, kinslaying & helcaraxe that she can smash him over the head with, figuratively speaking.
While we're at this, I remember from the session how the topic of Luthien's song explosiveness or not was discussed, I do agree that it shouldn't be a giant bang, but i think it should develop a lot of energy in a different way. I would love though if she starts out of the silence very low and almost (deceptively) weak, looking for some weakness because she recognises that Sauron would do the same if she goes for brute strength. which invites him investing more force and more info into his song as he thinks he sees weakness. Maybe she hears some kind of hint to the necromancy, and somehow teases out more info on that, which she then can use to judo-style sweep him off his feet. But i do think at the end she should be singing fully and with a lot of energy, once he is totally swept up in the song magic "flood" of it all. Otherwise that would be a little anticlimatic. But the force is not the thing that turns the tide. (Okay sorry enough confused water and sports metaphors for one paragraph xD)
