i know that (알아 = to know...) and (좋아하는 = "favorite/like"), but can someone explain to me all these words in that phrase context? I can't undestand the meaning of 날 and 거...
날 is a shortened term of 나를(나+를), meaning 'me'. 나 means I, and 를 is attached right after 나 in order to transform the meaning to 'me'. 거 and 것 is the same: roughly means 'thing'. And 을 is omitted - it should be right after 거(것) but there is nothing wrong with omitting 을 since 을 is a postpositional particle(조사) and omitting 조사 is grammatically allowed.
So, a more formal sentence would be "(나는 네가) 나를 좋아하는 것을 알아." which can be 'technically' and formally translated as "I know the thing(fact) that you like me." - leading to a bit more simplified term "I know you like me."