That’s clever! Do the tires help bump off pests?
No, not really; but they do provide early season heat.
They save space, and also make hilling & harvesting easy: as the vine grows, add another tire to the stack, and start filling it with straw. No soil pressure on the developing taters, so they form better.
When the vines die, unstack & remove the straw for the compost pile or for tilling under, and the potatoes are right there for the taking.
Other people build slat-sided boxes to do the same thing, adding 1X6 or 1X8 boards to the box frames as the vines grow.
The tires are also good for squash & tomatoes, mainly for the extra heat they provide.
Several gardening books cover it.
It also keeps the tires out of land fills; and now that we get charged “disposal fees” for leaving the old tires at the shop when they’re replaced, we keep them instead, and put them to use.