I bought a prepaid Visa gift card to add money on my phone account. It turns out I can't do this because the gift card can not be processed by my phone company because the card does not have my address to verify that it is truly me. Is it possible to transfer the money off the gift card onto one of my debit cards in order to pay my account?
Probably not. Gift cards are generally rigged up as payment-processing tools only, and you can't move the balance to a deposit account. It probably wouldn't be a horribly objectionable thing to do for the gift-card companies in general (heck, you've already paid a fee, why not let you have the money back?) It's just that they haven't bothered to set themselves up this way, since nobody does it.
You may be able to buy a money order of some sort with it (at a fee). Your best option is probably to use the gift card for everyday expenses, or perhaps run an errand for someone willing to pay you in cash (which you can then put in a deposit account).
I don't think the reason is "to verify that it is truly me". It should be possible for someone else (friend, relative...) to make a payment on your behalf, using their own card.
It is common that "gift cards" are not per-authorized for "Card Not Present" (CNP) use like on the internet, or over the phone. In many cases, you can register your card, online or by speaking to a representative over the phone. After that, you should be able to use it to pay your phone company.
Also, depending on where you got the card, you may be able to go to a teller at the issuing bank, and withdraw cash (or get a check), possibly without a fee.