The 1099-G reports tax refunds for taxes that were paid in previous years. The state doesn't know if you deserve a refund for 2015 until you file your tax return in early 2016. Therefore, even though the form says "2016" in the upper right, box 3 has "2015" because that was the year you originally paid the taxes that were refunded in 2016.
You should enter in TurboTax whatever year is in box 3 of the 1099-G.