basically, 뭐하고 놀아? here means "what are you up to?" or "what are you doing?" (its kinda tough to translate lol)
뭐 means what.
하고(하며) means doing/by doing ~ . also, 하며could be replaced for 하고
놀아 has multiple meanings 1.to play 2. to chill(hang out) 3.to de something fun . but in this sense, it's the thrid one
so it would be "what are you doing for something fun?" which could be translated to "watcha doin'??"
about ~고 could mean 'and' but as it merges with a verbe, it means 'by doing + verb'
lmk if you have something you don't understand or would like to know more about korean:) u can inbox me