You don't need ID -- they just ask you your name/address, check they have you registered, and check your signature against the one you gave when you registered to vote. If you're not registered you do need ID and proof of residence.
The current system gives you time to gather the required documents and recourse if the local government rejects it. It limits the ability of the people at the polling to turn you away.
It starts to make more sense when you consider all most people have are state issued IDs, combined with our political history of disenfranchisement -- particularly in the Jim Crow era.