It means to cause upset or disturbance.

If you upset the apple cart, you have changed the balance of things to make a situation worse. For example... you know that your best friend is cheating on his girlfriend. They are both happy. However, you manage to tell the girlfriend that he is seeing someone else and they have a huge fight. Maybe it was an accident but the situation has gone from calm and peaceful to angry and violent. You have upset the apple cart.