Solas attributes his vast knowledge on ancient elven culture to his travels in the Fade, and the things he's seen through memories relayed through Spirits beyond the Veil. He shares a great deal of this knowledge with the Inquisitor, whether or not the Inquisitor is elven (though some conversations need higher approval before being triggered for non-elven Inquisitors). He is particularly open for questions on topics regarding ancient elves, modern city elves, and Dalish elves—though he has very specific opinions on them. Keeping an open mind while discussing various topics and not letting popular biases or assumptions show will earn the most approval from him.
"The Dalish remember fragments of fragments, but that is more than most."
Besides his first personal quest involving activating left-over elven artifacts, he will notify players about nearby artifacts when he is in your party. These artifacts are wards meant to secure the area and make the Veil stronger and prevent demons from slipping through. Again, he shows great knowledge in how these work, and uses his knowledge to assist Lavellan and the Inquisition's efforts to seal rifts and heal the Veil.
Before the reveal, his knowledge seems like a typical video game gimmick (what better way to introduce the Orb Corypheus uses without asking Corypheus himself?). How many times do we meet characters in games that are used as a source of information and a way to expose players to the world of the games? At first it's not really easy to pick up that Solas knows a lot of ancient history for a little-known hedge mage, far outstripping the likelihood his journeys in the Fade could reasonably reveal. We take his knowledge for granted at face value, and he falls into the role of the Mentor archetype, guiding the Inquisitor to their goal.
Your other companions also fall into the same trap. Some are initially suspicious, but quickly grow to trust Solas's dependable knowledge; others point out the peculiarity in a humble mage knowing so much. Solas himself always plays these off, either by crediting his knowledge on experiencing the events through dreams in the Fade, or deflecting their praise by saying he's different because he's self-taught and their lack of mage training makes his difference appear more impressive.
"I don't know what to make of Solas. So much knowledge and so little personal history... I find that... peculiar, don't you?"
I'm biased, but I found his deflections to be awkwardly endearing. He lists off some impressive knowledge, only to quickly amend he had discovered them through his journeys in the Fade, not that he lived through any of it himself; that couldn't possibly be the case. He walks a very tricky line between being impressive with his vast knowledge but all the while appearing unassuming.
His character design was also very well-done. Almost all the other characters all have some impressive traits. The other mages, Dorian Pavus and Vivienne, both dress very flashy and do not miss opportunities to show off, but Solas remains modest. Once his true identity and purpose are revealed, it's clear his demeanor was meant to deflect suspicion by appearing very nondescript. Solas does not want anyone to look at himself or his past too closely.