In this phrase, "play" means a clever tactic or strategy.
In a game, you may have a choice of possible options. The options can be called "plays." For example, in a game of chess I might have the option of capturing a bishop with my knight now, or trying moving a pawn that might make it possible to capture the queen later. Capturing the bishop might not be my best play.
"Playing for time" means to do something to trick someone into waiting.
For example, a bank teller might say to a bank robber, "I only have a few hundred dollars here at the desk, just wait a minute while I open the safe... it didn't open... you've got me so nervous I'm having trouble remembering the combination, let me try that again... still didn't work... I just can't remember it, but I have it written down in a little notebook I keep in my purse, just wait a second and I'll get it..."