The non-resident state does not give you a credit for taxes paid to another state.
If a taxpayer lives in only one state, say Kentucky, and earns 40,000 in Kentucky, they pay only Kentucky state tax on that 40,000.
If a taxpayer lives in one state, say Kentucky, but earns 40,000 in say Missouri,
they would pay tax to Missouri on 40,000 income. They would also need to pay tax on 40,000 to Kentucky since most states tax you on income you earn in the state and also income you earn while a resident of the state.
So, first you file a non-resident state return for Missouri and pay tax on 40,000.
Next, you file a resident Kentucky state return.
When Kentucky goes to tax you on the 40,000, it says, "Wait, you already paid Missouri tax on that income, so we will give you a credit"
There is no credit on the Missouri return because the 40,000 had not yet been double-taxed.
The amount WITHHELD on your paychecks should not be confused with what the state tax LIABILITY will be. The state tax liability is determined on your state tax return, then, if the withholding was more, you get a refund. If the withholding was less, you have a state tax balance due.
@Tannerwalks
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