"~보고 싶어" is the casual conjugated way of the expression "~보고 싶다"
~보고 싶다 = Basic way (not conjugated) of the expression in english "to miss" (someone)
Casual (low respect):
~보고 싶어
Ex)
나는 (너를) 보고 싶어.
Formal/Polite (high respect):
~보고 싶어요
Ex)
저는 (당신을) 보고 싶어요.
Honorific:
~보고 싶습니다
저는 (당신을) 보고 싶습니다.
In fact there are 5 ways to talk politely (or not) in Korean, but these are the most common ones.
I hope I'd helped you! ^^ good luck with your learning!