Why did landing a rocket upright change the cost of spaceflight?

Because, by SpaceX's own figures, a rocket's fuel is the cheap part and the hardware is the expensive part. In 2014 Elon Musk said the propellant for a Falcon 9 is only 0.3 percent of the total cost of the vehicle, and that success with reuse could reduce the cost of spaceflight by a factor of 100. That was a goal, not a result. A reusable launch vehicle saves building new parts for each launch, but the benefit can be eaten up by the cost of recovery and refurbishment.

Reuse is hard because so little of a rocket reaches orbit: a payload is typically about 3% of the rocket's mass, roughly the amount of fuel needed for re-entry. A reusable Falcon 9 first stage has to carry extra propellant and landing gear, which means around a 30 percent reduction of the maximum payload compared with the expendable version. So every landing has a price in payload.

The landing itself is a sequence: after stage separation the booster flips around, a boostback burn can reverse its course, a reentry burn sheds speed to prevent overheating, and a landing burn does the final deceleration and touchdown. Grid fins added from 2014 steer it to a precise landing. SpaceX first landed an orbital rocket stage vertically on 21 December 2015, and first reflew a first stage on 30 March 2017.

Is it actually cheaper? In 2015 industry analysts warned that refurbishment and relaunch costs were not yet demonstrated, and that the economic case would depend heavily on launching frequently. In 2016 SpaceX's president said US$1 million of refuelling and US$3 million of refurbishing could let a reused launch be priced as low as US$40 million, a 30% saving, and in 2020 Musk tweeted that reuse costs less than 10% of a new booster (these are the company's own figures).