Okay kiddo, imagine you and your friend Johnny are playing a game of tag. Johnny is "it" and you are running away from him. Suddenly, Johnny tags you and you become "it" too.
So, in the situation where you were tagged, Johnny is the person who did the tagging or the "art" of the situation. You, on the other hand, are the person who got tagged or the "part" of the situation.
Now, let's say that you and Johnny were caught by your parents and they want to know who was responsible for the game. If Johnny tells them that you were the one who came up with the idea to play tag, then he is the "part" of the situation because he was just following your lead or your "art".
In summary, "art and part" is a phrase used to describe two different roles in a situation - the person who comes up with the idea or does the action (the art), and the person who participates in the action or is affected by it (the part).