There are many arguments for allowing convicted (but released) felons to vote. It must be remembered, however, that this is a decision made at the state level, so different states have different policies. Two states (Maine and Vermont) actually allow prison inmates to vote.
One argument is fairly simple. A person who is convicted of a crime receives punishment by being sent to prison. Their prison sentence is their punishment, and, theoretically, intended to rehabilitate them. If they are released from prison, having "done their time," they should be able to rejoin society as productive citizens, and this includes having the right to vote.
Another argument is that people who have been convicted of a crime are still represented by elected politicians. The policies enacted by these politicians still affect them. Therefore they should be, under the principle of representative democracy, able to vote for or against people in office. To deny them this right is to strip them of the right of citizenship for life.
Another potent argument stems from the reality that African American males are disproportionately represented among people convicted of a felony. States that strip convicted felons of the right to vote disfranchise huge swaths of their African American populations. This is an especially urgent issue in southern states. An advocacy group recently determined that over twenty percent of the black population of several southern states lacked the right to vote for this reason.
Challenges to state policies that deny the vote to former felons point to the Constitution. Some have charged that these laws violate the Equal Protection Clause of the Fourteenth Amendment, while others argue that it violates the spirit of the Eighth Amendment, which guarantees protection from "cruel and unusual punishment" and "excessive bail."