At the beginning of Harry Potter and the Deathly Hallows, Snape is cleanly out of the order. The secret of when Harry was moving was very closely kept. It seems even the staunchest Order members don't trust him. So, how did he figure out when Harry was going to be moved to the Weasley's home?
If Snape could Confund Mundungus Fletcher to suggest the plan of using multiple Potters for Harry's move, then no doubt he could in the similar fashion get the date from Fletcher after it was set. Snape, highly accomplished in Occlumency, was probably a strong Legilimens as well and may simply have read it in Fletcher's mind. It isn't spelled out in the final book, but that seems the most likely way.
It's possible that one of Dumbledore's reasons for having Fletcher -- most likely not all that trusted by the Order himself once Dumbledore was no longer around -- suggest the plan of duplicate Harrys was to ensure that Fletcher would know the date of the move, making it easier for Snape to get when it was finalized.
A possible alternative is that Dumbledore's portrait got the information from Phineas Nigellus's portrait. But that would have required the plan to get finalized at Grimmauld Place, which the Order pretty much abandoned after Snape killed Dumbledore, and within earshot of the portrait.