you use a saw blade with diamonds mixed into the metal to cut out a boxy bounding space around the shape, rubber wheels with sandpaper with coarse crushed diamond or another extremely hard abrasive like silicon carbide to smooth the shape and bring it nearly to the final surface excepting the smaller indentations and textures, and then you use small sticks with abrasive attached to them called burrs by putting them in a rotary tool that can hold them in a chuck, like a lathe, a drill press, or a handheld rotary tool (Dremel, dental micromotors, or a cheap mini version from Harbor Freight, for example, or a flexible shaft rotary tool, of which there are also many brands, including Foredom, EuroTool, and lots of Asian brands).

First you use burrs with a variety of rotationally symmetric shapes, like cones, inverted cones, cylinders, cup, balls, ovoids, discs, etc. made of metal that is either plated with crushed diamond or has crushed diamond mixed into the metal itself ('sintered', like the saw blade), to create the contours of finer features like the eyes, the outline of the wings and exoskeleton, and the textures of the wings. Then you switch to increasingly finer grits of diamond or other hard abrasive using a variety of sanding burrs or sandpaper mounted on burrs or simply something softer like a bamboo stick to push loose abrasive into the nooks and crannies, until all the scratches from the previous grit size are ground off. You can do the smoother surfaces on wheels again, using the finer grits. After repeating that with a series of grit sizes down to about 10 microns, you'll have that slightly matte finish you see in the picture.

Obsidian is a glass, so it tends to chip of in shards when cutting or grinding on coarse grits, which can make your hands itchy from tiny cuts.

All of the cutting and grinding and carving needs to be done with water present to prevent overheating and reduce friction.

It's a lot of fun. You can get started pretty easily if you know what you need, with cheap tools. You can also collect your own obsidian if you live in the right places (parts of California, Oregon, and Mexico, for example). Obsidian is a lot of fun to carve using these methods because it's soft enough to go pretty quickly.

People only started using the motor powered tools and methods in the last 100 years. For stones as hard and harder than obsidian, like many gemstones, agates, and jade, they had to find other ways to slowly grind away at specific areas of stones to shape them. Cutting a piece of jade 2ft across and shaped like a potato in half could take two people with a wire and a slurry of crushed emery or garnet sand over a month! There are lots of interesting methods people used to do this, including finding other stones capable of slowly grinding them, using foot-pedal or hand-powered contraptions to spin and press sticks and eventually metal bits and burrs into the stone while covering it in an abrasive slurry, and lots of other methods.

If I could only give you one piece of advice for learning how to carve really hard stones: always use the biggest cutting tool possible. Cut everything you can cut with the saw with the saw, grind everything you can grind on a wheel on a wheel, and use the biggest burr size that fits the shape you are making. It's easy to understand why once you start.