Here's the best explanation of this I have seen, from the Lady Bird Johnson Wildflower Center:
You take some rose cuttings, put an intact, raw potato on one end of the cutting, and stick the whole thing (potato end first) into the ground. Apparently, the moisture and nutrients in the potato keep the cutting end moist and give it enough of a mild "fertilizer" that it helps the cuttings take root. When the potato has been used up, the rose cutting is rooted. Voila! Haven't tried it myself, but it looks dead easy. Good luck!
The downside is that you may end up with potatoes instead of roses:
We tried this and all we got were potato plants! A waste of time.