Not sure how it works in other states, but in mine, it is basically a consumption tax (equal to the states sales tax) on any items brought into the state from a jurisdiction with no sales tax, or with a lower sales tax than Ohio. In order to comply, you would need to keep track of every item bought in another jurisdiction and the amount of sales tax paid for each item. If you paid less than 5.5% (plus your county's additional, up to 7.5%), you owe the difference.

Strangely, there doesn't seem to be a credit for items purchased in jurisdictions with higher sales taxes.