I assume that you have a mutual fund. Generally the fiduciary will include with the 1099-INT the information you need. Lacking that, you can find the information online.
Most mutual funds will provide a breakdown. But you usually have to ask for it, or find it on their web site.
If they did not provide a breakdown, you check the box “I earned tax exempt dividends in more than one state” ("Multiple States" in the online program) on the first screen after entering the 1099-INT or 1099-DIV. Then select "More than one state" at the bottom of the state scroll down list.
If your mutual fund company provided you a breakdown*, you are only interested in your home state*. Multiply the % for your state by your total tax exempt dividends to get a $ amount (you can't enter the % in TurboTax [TT]). When asked which state, check the box "I earned tax exempt dividends in more than one state". In the drop down menu, select your state and enter the $ amount you calculated. In the 2nd box, select "More than one state*" (at the bottom of the scroll down list) and enter the remaining dollar amount.
*Most mutual funds will provide a breakdown. But you usually have to ask for it, or find it on their web site.
**Your state will tax all the dividends except the dividends from municipal bonds from your state and US Territories. In Illinois, only certain types of bonds are state tax-exempt.