I hate to say it but what kind of taxi clearance is "UAL1544, taxi to runway 9L, exit the ramp at Tango 7."?
I've never seen that kind of phraseology in the 7110.65. I wonder if the by-the-book phraseology, "Taxi to Runway 9L via Tango, Tango 7, Delta, Bravo," would have prevented the runway incursion? Even a "Exit the ramp at Tango 7, right Delta, Left Bravo" would have been more helpful. The ground controller was a trainee being supervised by the female controller at the beginning of the clip. Again, the enter and exit the ramp stuff with no mention of the taxiway routing TO the assigned taxiway gets her in trouble in the beginning, where there seems to be a mis-communication with a foreign pilot Avianca 036 who just exited Rwy 9L and asks for a clarification and another routing. Her clearance is "Avianca 036 enter at Tango 4 [being the taxiway] taxi to the ramp, hold short of Tango 8 [there is some ambiguity here as to whether T8 is a taxiway or gate - it's a taxiway] traffic's pushing off that gate now." and then "Enter at your next left hold short of Tango 8." But Tango is a taxiway and thus a movement area that is adjacent to, but not "the ramp" (a non-movement area). She could also mean Tango is the "Inner," but doesn't specify so, and wouldn't run into that problem if she just called it Tango to begin with!
I don't mean to split hairs or insinuate that controllers should hold the pilot's hands and progressively issue every taxi instruction at every airport, but FLL has intersecting runways (one is active and one of which isn't, and is being used for taxiing) and the taxi route to the active runway crosses the inactive one (need to cross 13-31 to get to 9L). In this situation, the pilots are thinking about the part of the AIM that states that "Taxi to runway 9L" implies that the pilot may cross all runways except the assigned runway. Now they're anticipating that they'll pass the red runway boundary signage and cross the hold short lines onto runway 13-31 WITHOUT CLEARANCE (since none is necessary) on the way to their assigned runway. With that forethought in mind, and a vague clearance that doesn't specify which taxiways to take after Tango 7 (or remind the pilots that there will be a LEFT turn onto bravo), one can begin to understand why they missed it.
It's not that they were idiots and didn't know airport signage and how to hold short of a runway, they were actually anticipating the runway crossing, got distracted by something (presumably thought they turned), didn't read the numbers on the signs closely enough and crossed the WRONG runway. I hope Rwy 9L at Delta got marked as an incursion hot spot, and that training at FLL ATCT has changed to promote more detailed taxi instructions in that vicinity of the airport to alleviate future incidents like this one. But hey, just one man's opinion.