Because felons are the BAD PEOPLE and BAD PEOPLE shouldnt be allowed to do anything.

The real reason felons are barred from voting is that its a really easy way to suppress a group you don't like. Just police a group more intensely for things like drug possession or petty thievery and suddenly you're getting rid of people who might vote against you. Weed related offences being a felony was a big cause of disenfranchisement among hippies and is still a problem among people of color.

I decided I'd look it up: Time covered the rational in one of the states where you cannot vote if you are a felon, Mississippi, by breaking federal law you demonstrate poor judgment and that after doing time in prison you'll likely be estranged from the political parties.
Personally I don't agree with extending a punishment to someone after they served their sentence, (I suppose I hesitently agree with felons not being permitted to own/possess firearms though;) if former prisoners are really such a hazard to society then why even let them out?
To me it's another failure of punitive prisons all the more reason our prison system should focus more on reform.