I don't believe there is any technical innovation that is responsible. After all, all the other competitors have access to the same technologies, but they aren't doing it.
I would argue the technology was there before:
- Blue Origin also autonomously lands their New Shepard booster, and even did it successfully before SpaceX did (although, of course, New Shepard is not an orbital system, it just goes straight up and down again).
- The DC-X demonstrated autonomous vertical landing in 1993 (it was intended to land from orbit, but all demonstrations were just hops of at most ~3km; on its penultimate flight, it even demonstrated the Starship-style reorientation from belly-down to landing feets-down).
- Autonomous landing from orbit (but not vertical) was demonstrated with Buran in 1988.
- Autonomous landing of an aircraft was demonstrated in 1945.
The innovation is a commercial one. It is the simple realization that it could actually make sense to do this.
Before SpaceX, nobody was doing it, because nobody saw the need to do it: why reuse a rocket when you can simply charge the government 10 times as much money to build a new one every time? It was the commercial realization that there are organizations without an infinite supply of money that also want to launch stuff into space, and the commercial realization that there is enough demand for launching stuff into space that building rockets fast enough would become a problem, which led SpaceX to focus on reusability.
Note that 37% of all SpaceX Falcon 9 launches were for a single customer. And that customer is SpaceX itself! If we exclude the first 70 launches and only focus on the launches since Starlink actually exists, that number goes up to over 50%. More than half of all Falcon 9 launches since the first operational Starlink launch have been Starlink launches.
Or, looking at it from the payload perspective instead of the launch system perspective: 50% of all satellites launched by all countries in the entire history of spaceflight are Starlink satellites. That means, SpaceX has launched as many satellites just for Starlink in 4 years as all of humanity combined in 70 years.
Now, all of a sudden, cost becomes very important because you don't have a customer paying for the rocket, you yourself are paying for the rocket. And if you are launching thousands of satellites, you have to launch very often, very fast, otherwise it takes decades to launch your constellation, so you simply do not have the time to build rockets fast enough. Now, reuse not only makes sense but becomes a necessity.
But Starlink is a customer no other company has. So, no other company (so far) has seen such a pressing need for reusability.
I don't believe it is a coincidence that among the other companies working on reusability, some of the ones pushing hardest, are Rocket Lab and Blue Origin. Rocket Lab also designs, builds, and runs satellites, and offers a full end-to-end mission service package. And Blue Origin is owned by Jeff Bezos, who also owns Amazon, which is planning to launch their own Internet mega-constellation.
There is one technology that helped especially with the drone ship landings, though: the ubiquitous availability of high-precision, high-accuracy, realtime position information through GNSS systems like GPS, GLONASS, Galileo, and Beidou, especially combined with augmentation systems like dGPS and SBAS (e.g. WADGPS, WAAS, NDGPS).
As mentioned above, the first autonomous landing of an aircraft was achieved in 1945 using technology that later became the internationally standardized Instrument Landing System that is still in use today. While it would be possible to use a similar system for rocket landings, it would require some large antennas and power hungry transmitters on the drone ships. GNSS definitely helps here, but it is not strictly necessary, especially for land landings.
It is quite ironic that most of SpaceX's "old space" competitors like Boeing, Lockheed-Martin, Northrop-Grumman and co. are not only launch system providers but also weapons manufacturers and defense contractors as well as spacecraft designers. All of them have already demonstrated precision guidance in hypersonic, supersonic, transsonic, and subsonic regimes using various aerodynamic control surfaces such as fins, speed brakes, and grid fins. Both missiles and precision bombs, for example, use grid fins. Several of them have built VTOVL or STOVL aircraft that perform jet-powered vertical landing (which is not too dissimilar from rocket-powered vertical landing). All of them have autonomous drones. Several of them have demonstrated autonomous precision landing of spacecraft on other celestial bodies.
Hitting a target with high accuracy is quite literally their business.
Furthermore, Boeing is the current owner of McDonnell-Douglas, the company which already demonstrated a VTOVL orbital vehicle in 1993. Boeing is also the current owner of Rockwell, which built the Space Shuttle Orbiter.
And yet, Boeing did not implement reusability either in SLS or in Vulcan. And why would they? For SLS, their customer seems to be quite happy to pay \$1–\$4 billion (depending on estimate) per launch, so why make the rocket cheaper through reuse if that only means you get less money?
ULA only seriously started to consider partial reuse of Vulcan after they got awarded the 38-launch mega-contract for Kuiper. However, their reuse concept does not involve landing; instead, the engine section and thrust structure which also houses the first stage avionics, detaches from the booster and reenters using an inflatable aeroshell. This concept had been presented for 10 years, but it only existed in PowerPoint and work didn't start until after the Kuiper contract.