They tagged the missile.
Keeping enough laser on it steadily to burn it up, despite intervening clouds and any reflective/refractory coating on the missile itself, is another thing.
It doesn’t take long to burn through the very thin-skinned missiles and destroy their aerodynamic integrity/warhead controls/flight controls and electronics.
“Dumb” single-stage/gravity-ballistic missiles like the Palistinian’s Katuska’s have also been routinely destroyed since the early 2000’s by the THEL and other laser versions.
The lasers are actually chemically-powered by some “complex” H2OH (etc) reactions. You don’t use electric power to create the beam, just mix the chemicals. Given enough liquid in the 747 tanks, it means you can’t run out of “anti-missile missiles” .
While not mid-air refueled now, such a reload capability could be provided by specialist tankers.