This is the 1151st most frequent Korean word.
보여
"보여" means "shows" or "is visible."
In this sentence, '보이는' derives from '보이다,' meaning 'to be seen' or 'to look.' Here, it expresses 'look tired' as '피곤해 보이다.'
그가 얼마나 피곤해 보이는지 알 수 있었어요.
I could tell how tired he looked.
Here, '보여요' comes from '보이다,' and it indicates how something appears to the observer, in this case, '아름답게 보이다' or 'seen as beautiful.'
그녀의 미소는 정말 아름다워 보여요.
Her smile looks really beautiful.
In this context, '보여요' is used to express the perception or how something (제 제안) appears to someone, combining with '이상하게' which modifies the verb.