보고 싶어요 is the 어 conjugation of the verb 보고 싶다, meaning “to want to see.” This verb comes from 보다, meaning “to see.”

To review the 싶다 ending, see my lesson on how to say “I want to…” in Korean.

보고 싶어요 and 보고 싶어 have become strong shows of affection, stronger than that between casual friends. They’re often used in romantic situations.