What do you mean by re-directing state taxes owed?

Making a donation to APESF is an alternative to writing a check to the state. For example, if you owe $3,000 to the state, you could say, “I want $1,000 of that to go to students in Arizona private schools,” and you would then only owe $2,000 to the state. In other words, a tax credit contribution to APESF directly reduces the amount of taxes you owe to the state of Arizona that you would have to pay anyway. By taking the credit you are, in effect, re-directing your tax dollars to benefit worthy students in Arizona private schools.