Resizing feels like magic to a lot of customers, but it's really just careful, precise metalwork. Here's what's actually happening on the bench.
For most rings, your jeweler cuts the shank at the bottom, opposite the stone. To size a ring down, they remove a small sliver of metal and solder the two ends back together. To size up, they add a matching piece of metal into that same cut and solder it in. Either way, the ring is then reshaped on a steel mandrel, the solder joint is filed smooth, and the whole band is re-polished so the seam disappears.
A few things I always flag to customers. First, resizing has limits. Most rings can safely move about two sizes in either direction before the structural integrity or the proportions start to suffer. Second, eternity bands, rings with stones set all the way around the shank, or heavily engraved bands, are much harder to resize because there's no clean, stone-free spot to cut. Sometimes those styles need a different approach entirely, or simply can't be resized. Third, sizing up thins the metal slightly at the seam, so a ring that's already delicate may need reinforcing.
One tip from years of watching this go wrong: don't get sized right after a big life change like surgery, a new medication, or extreme weather, since finger size can shift temporarily. A good jeweler will ask you a few questions before they ever pick up a saw.