친해지고 싶어요(chinhaejigo shipoyo)>> "I want to be friends with you." ^^ verb+고 싶다 means "want to (do something)" so yeah, do you know 친구? it means friend(s), right? so by looking at the sentence structure (including the words), most probably, one would assume someone is asking/offering friendship. ^^

Anytime you see -고싶어요, you know it means "I want to...". 친해지 translates to "friends" or "intimacy", so I'm assuming it's like "to be friends". So, altogether, I would say that translates to "I want to be friends." note: I am just learning Korean myself and did this as an exercise. I would like to have a native speaker answer too.