There's four main ways usually, steam bending, kerf cutting, carving, and faking it.

With steam bending you get a bunch of thin pieces of wood, steam them, and then put them in a mold. They dry and hold the shape of the mold. They're glued together to be a strong curved plywood basically. If I had to guess without seeing it. This was the method used.

With kerf cutting you cut a bunch of parallel lines across the back of the area you want to bend. Just the thickness of your saw blade. The front of the wood is left thick enough to hide the cuts, but thin enough that, often with some steam, you can get the wood to bend closing the gaps cut in the back.

The third way, carving, means you start with a large piece of wood and just cut a curved section out. This is the least common approach as it's hard, wasteful, grain doesn't always run in a strong direction, and often impractical to find wood large enough.

Finally, faking it. Usually gluing veneer (a very thin sheet of wood) over something. Think laminate countertops, but the reverse. More and more it's by just printing on a wood pattern scanned from actual wood or something.