The taxes paid in NY are often higher than the taxes paid in NJ. NJ will give you a credit up to the amount of tax that you would pay in NJ on the same income. On the first screen you may see the tax that you paid in NY, and then on the second screen you'd see the credit that NJ is providing. The credit in NJ is usually lower than the amount of tax paid in NY.
If you want to see the breakdown of your credit for taxes paid to New York, take a look at your Sch NJ-COJ. You should be able to verify Line A (the income taxed by NJ and NY), as well as Line B (your NY tax which should agree to your IT-203, Line 50). The remainder of the form are calculations to come up with the amount of credit allowed for NJ.
If you are using TurboTax Online, click here for instructions to view your forms.
If you are using TurboTax Desktop, simply go into Forms (top right corner of your screen) and look at the form.