You need to have the Nursery window open to sell you plants. If you change game screens, change user/slot, or exit the game, the customers won't buy your plants.

If you are running in windowed mode on a PC, you can keep the nursery window behind other applications, and the plants will sell (I'm doing that right now as I type this reply!)

However on Mac OSs this doesn't work, and the focus needs to be on the game window.