When you press the play button on the playback option of Plane Finder, all of the planes which are flying in real-time should disappear for a moment momentarily and then after a short while, some planes will appear back on your screen. These are the aircraft that you have selected to show on your screen at the selected time by your playback time slider.
The most obvious reason is probably of a slow internet connection and this delays the time it takes for planes to show up on your screen after the planes have momentarily disappeared. How fast is your internet connection and how long does it usually take for the website to fully load up with the aircraft displayed on the screen? I would suggest waiting for a few hours and then trying again. If this does not work, then select the playback time and wait for perhaps 1 minute to see if there is any change.