You will need to file a state tax return for Connecticut since you are a resident.
CT residents are taxed on all their income, including income they earn in a different state.
Connecticut also give a credit for taxes paid to another state. So although you will be claiming the same income twice, once for the state you live in and once for the state you work in, you will get a credit on your Connecticut return for the tax you pay to Vermont.
Do the Vermont return first because the credit is not based on what was withheld, but on the final tax liability calculated on the Vermont return.
Next, do the Connecticut return. The Connecticut tax and credit will be calculated on that return.
Vermont has a higher tax rate, so the credit should zero-out any tax due (on that same income), or at least come pretty close.
**Say "Thanks" by clicking the thumb icon in a post
**Mark the post that answers your question by clicking on "Mark as Best Answer"
