Balsamic somehow makes tomatoes taste more tomato-y. With its rich, caramel-like flavor, balsamic balances the too-bright flavor that tomatoes can sometimes have and helps the other flavors in the dish come a little more forward.
Just a splash is all you need for a batch of gazpacho or a pan of tomato sauce. No more than a tablespoon. Balsamic is the side-kick here, remember, so you don't want to add so much that it overwhelms the tomato.
Ideally, you'd never now it was there if you didn't watch yourself pour it in; you'd just know that your soup tastes infinitely better.
Do you ever do this?