"The man laughs with the woman."
Translation:남자가 여자와 같이 웃습니다.
That would translate to "The man and the woman laugh" : subject being "The man and the woman"
Here the subject is only "The man", who is laughing with the woman. Therefore 남자 receives the subject marker and 여자 receives the conjunction marking that the subject is doing the action with her.
From what I understand if the sentence was 'The man and the woman laugh together' then it could be written as 님자와 여자가 함께 웃습니다. But since the original sentence is The man (the subject of the sentence) laughs with the woman, the correct way to translate it is 님자가 (가 to indicate the man is the subject) 여자와 (와 - with) 함께 (that they are doing said action together) 웃습니다. I could be wrong tho, since I am a learner myself. But this seems like a logical explanation.