한국남자친구 --> Korean ("한국") boyfriend ("남자친구")
남자 = man, 친구 = friend. slang for boyfriend is also "남친"
자랑하시네 --> is bragging about.
하시네 is a present-tense, declarative, neutral honorific form of the verb "하다" and 자랑하다 means to brag or to boast or to tell proudly, etc.
the whole phrase can be "brags about a Korean boyfriend."
without more context it's difficult to say what the exact situation is or what the exact translation into English would look like :)