Hi It means he smiled at me Keep it up with your english!

He gave me a big smile = He smiled at me: 나에게 큰 미소를 지었다. He made me smile = 날 웃게 만들었다.

"he gave me a BIG smile"= he smiled at you with A BIG Smile, the smile was warm and obvious, .. Big is being used as an adjective to describe the smile being warm enthusiastic and joyous. "he made me smile" = he did or said something to make you smile. He made you smile.

"He made me smile"means that someone made you happy by telling joke on the other hand "HE smiled at me" means that he One who derides, or laughs at, another in contempt; a mocker; a scoffer.

"He gave me a big smile" = "He smiled at me" "I gave him a big smile" = "I smiled at him"