How did old laws against teaching slaves to read and write make difference after the civil was?

I think you are confused. It was against the law to teach a slave to read and write. After the civil war exslaves weren't allowed to attend schools and the Jim Crow laws made sure that African American children weren't allowed in the white school systems. The idea of "separate but equal " was encoded for a 100 years into the educational system of the south. Thus, the literacy tests proved successful because to register to vote required reading and answering questions on the constitution.