What does “Ground loops are costly. Fly them until they stop” mean?
It just means "pay attention to what you are doing" until you are no longer moving.  "Fly them" means "keep actively controlling the plane as if you were still in the air". Don't start daydreaming just because you're on the ground, in other words.

Especially with the taildraggers of olden times; 


They are dynamically unstable turning-wise while rolling along the ground (because the CG is behind the fixed traction points; try driving a car backward really fast) and require careful attention to staying directionally aligned even while moving slowly.  Taildraggers with locking/free swivelling tailwheels (which you tend to see on the larger ones) are even more demanding, and are basically like a shopping cart rolling backwards (the kind with fixed back wheels, not the 4-wheel caster kind) once the tailwheel is unlocked for taxiing, and need to be steered with brakes alone, which requires quite a bit of concentration.
They like to tip forward under hard braking so keeping full up elevator while taxiing is a requirement to minimize this tendency
Strong quartering winds, if present, are trying to tip you over while 3 point on the ground and you have to learn to apply appropriate amounts of aileron to counteract this tendency.


A lot of stuff to keep in mind.  The tendency of a new pilot, when they had a successful landing, was to breath a sigh of relief and relax during the rollout when dropping below, say, 30 kt where rudder power fades away, or during taxi, forget they were in a backwards rolling shopping cart that is trying to switch ends continuously and let their mind wander, and oops.

Tricycle gear airplanes are nowhere near as physically demanding, but still you need to pay attention to what you are doing so it still applies in the end.