This is the 5476th most frequent Japanese word.
うかがえる
"うかがえる" means "can be observed, inferred, or discerned."
In this sentence, 'うかがえる' is used to express that something (his thoughts) can be conjectured or inferred based on evidence (this document).
この資料から彼の考えをうかがえる。
From this document, his thoughts can be inferred.
Here, 'うかがえる' conveys that the feelings of the subject (her) can be understood or perceived indirectly through her words.
彼女の言葉から彼女の気持ちがうかがえる。
From her words, her feelings can be perceived.
In this usage, 'うかがえる' indicates that someone's state (being troubled) can be interpreted or deduced from their facial expression.