Mass comes in several classes: metre- to kilometre-sized planetesimals; mm and cm sized dust and pebbles; and gas. The primitive solar nebula is overflowing with gas, has about 1% dust, and has a sprinkling of planetesimals.
When the primitive cores are barely more than wee planetesimals themselves, they accrete pebbles. Eventually they get big enough to sweep their orbit clear and keep it clear, and it this point they can't accrete any more pebbles ("the pebble isolation mass") and stop growing - except for increasingly infrequent collisions with planetesimals, comets and asteroids. That's what happened to the outer "ice" giants.
But Saturn and Jupiter were able to successfully accrete gas and balloon up in size. It's this phase I suspect they're talking about. And I suspect it's routine hydrostatic equilibrium: the outwards pressure of hot gas, retards gas from being accreted. Eventually, after a few megayears, the gas envelope has cooled sufficiently and the planet increased in size to a point where it's gravitational attraction can overcome the thermal pressure and it wraps itself in gas like a stick wrapping itself in candyfloss.