The wikipedia entry mentions that they might have left, maybe to try and attack the rear, but this seems like a terrible mistake by the Persians. Presumably they would have recognized that the Greeks were vulnerable to cavalry?
Krenz in The Battle of Marathon (2010) suggests the reason for Militiades' decision to charge was to catch the Persian cavalry before it could deploy. By attacking so quickly, it bottled up the cavalry behind the Persian infantry and left it no space to deploy on the battlefield.