The short answer
A polish with a mild abrasive removes them by taking off a very thin layer of clear coat. If your fingernail catches in a scratch, it is too deep for polish.
- Test the depth. Run a fingernail across it. If the nail catches, the scratch is through the clear coat and polishing will not remove it.
- Wash and dry the car thoroughly first.
- Use a mild polish on a clean applicator, working a small area at a time.
- Work in straight overlapping passes, not circles.
- Wipe off with a clean microfibre and inspect in strong light.
- Protect afterwards with wax or a sealant, because polishing removes whatever was there.
What will ruin it
Polish removes material and the clear coat is only a fraction of a millimetre thick. It can be done a limited number of times over a car's life, and machine polishing in inexperienced hands can go through it in seconds on an edge.
Why it works
A swirl mark is a groove that scatters light. Polishing does not fill it, it levels the surrounding surface down to the bottom of the groove, which is why it is finite.
Household chemicals can be dangerous in combination. Never mix them, keep the room ventilated, and test anything on a hidden patch first. Full disclaimer.