You do pay taxes to Massachusetts on your income earned there. With state taxation, your resident state (Rhode Island) taxes all of your income no matter where you earn it. Your nonresident state taxes any income you earn in that state. In some cases, states have a reciprocal agreement that allows you to be taxed only in the state you live. If RI and MA had a reciprocal agreement, then you would get back all of the MA withholdings. However, they do not. You must file tax returns to both states on the same income, also called double-taxed income.
With this arrangement, your resident state will grant you a credit for the amount of tax that the nonresident state taxes the income earned there. You will prepare the Massachusetts return first, so that TurboTax can calculate this credit on your Rhode Island return.
Please see this FAQ with additional information: https://ttlc.intuit.com/replies/4777207