Click keys to play notes. Hold down Shift for pedal. If your keyboard is set to QWERTY, you may use your computer keyboard: the middle row (ASDF...) for the white keys and the top row (QWER...) for the black keys. To change the range that the computer keyboard uses, option-click the note you want the "A" key to give, or use + and - to change the range by an octave. (Make sure to click inside the applet before doing anything using the keyboard—one of the keys should be orange if you can use the keyboard.)
If pressing and holding a key on your keyboard doesn't hold the note properly, from what I can tell that's a problem with the Java plug-in (which older versions of the plug-in and/or browsers didn't have). Click on the applet and press up to open the keyboard in a new window; this should fix the problem. (Also, it doesn't support pressing multiple keys at once, no matter what.)
(Also, when using the windowed version, or when running as an application, you can press Down, and it will show all the notes you play (but not the rhythms) in LilyPond notation. Click the text field and press return to play back these notes.)