Ah, the ESV's breaking up of the runon sentence I think makes it harder to catch the meaning. Because they disconnected "He listened to Paul speaking" from "And Paul, looking intently at him" with a period, I completely missed that all this happened while Paul was speaking. Until I went to Unbound Bible to look at the Greek, and I also read the ASV.
This man was listening to Paul as he spoke, who, when he had fixed his gaze on him and had seen that he had faith to be made well, (verse 9, ASV)
ουτος ηκουσεν του παυλου λαλουντος ος ατενισας αυτω και ιδων οτι πιστιν εχει του σωθηναι (verse 9, Robinson-Pierpont Byzantine Majority Text, 2000 edition)
The actual grammar in the Greek is just like the ASV has it. Unlike the ESV's "He listened to Paul speaking. And Paul, looking" it is a runon, with Paul's name occuring only once, and with the word ος (who) separating two clauses.
The word "seen" is just ιδων, which is common for actual sight as well as perception. I don't see anything special to take note of as far as Greek terms involved. But the runon sentence aspect of this makes it more dynamic than when its broken up into mini-sentences. It kind of evokes the idea of Paul preaching away, while this guy is in the audience nodding his head or in some way making it obvious that he believes what Paul is preaching, and Paul seeing this perceives that he has faith.