When someone goes apeshit, they suddenly explode with anger — shouting, slamming things, completely losing the calm face they had a second before. It is a dirty, spoken word. You will meet it in films, games, and angry talk online; you will not meet it in a textbook, a job interview, or a polite message to a teacher. The usual shape is go apeshit (at someone, or over something), not apeshit used alone as a quiet describing word. People also stretch it: go apeshit can mean go wild with excitement, not only with rage — a crowd going apeshit at a concert is loud and crazy, not necessarily furious. Though actually, the ape picture is just a cartoon of a wild animal; it is not a science lesson about real apes. If you need the same idea in decent English, say got really angry or lost their temper.