Many people think Elon Musk catches entire spaceships from the sky, but this isn't quite accurate. SpaceX actually focuses on recovering specific rocket components, not whole spacecraft returning from orbit. The main recovery methods involve rocket boosters that land themselves and fairings that were once caught with nets.
The most impressive recovery method is the Falcon 9 booster landing. After separating from the second stage, the first-stage booster flips around, fires its engines to control its descent, and lands vertically either on a drone ship at sea or back on a landing pad on land. This isn't catching from the sky, but rather the rocket flying itself back home.
The closest thing to actually catching from the sky was SpaceX's fairing recovery program. Fairings are the protective nose cone halves that shield the payload during launch. SpaceX used to deploy ships with giant nets to catch these fairings as they parachuted back to Earth. This was the literal sky catching method, though they now primarily recover fairings from the ocean after splashdown.
What about spacecraft that actually return from orbit, like Crew Dragon capsules? These don't get caught from the sky either. Crew Dragon capsules returning from the International Space Station perform ocean splashdowns using parachutes, similar to the Apollo missions. Future Starship vehicles are planned to land propulsively like the Falcon 9 boosters, not through any aerial catching mechanism.
To summarize: SpaceX doesn't actually catch entire spaceships from the sky. Rocket boosters perform controlled propulsive landings, while spacecraft like Crew Dragon splash down in the ocean. The fairing net-catching method was the closest thing to sky catching, though it's less common now. The real innovation is in making rockets reusable through precise landing technology.