The mid-air collision occurred because the Russian air crew maneuvered contrary to their TCAS instruction (it commanded them to climb, the controller ordered them to descend). They were not trained that TCAS is the ultimate authority in this situation; it exists precisely because the controller has already failed in their separation duties, and if you have TCAS giving you a resolution advisory, your aircraft is no longer under ATC control and you must ignore any ATC instruction to the contrary. The other aircraft was correctly following its TCAS instruction (descending) because their crew was trained in this. Both planes descended and still hit each other.

In this case, KLGA has RWSLs (Runway Status Lights), including RELs (Runway Entrance Lights) on taxiways, that behave like traffic lights on roads. This too is completely automated and is the last-ditch resort for when a controller has already failed in their separation duties. This system processes transponder data of nearby aircraft and determines whether an aircraft is about to take off (is on the runway and accelerating) or land (is approaching the runway and descending). In either case the RELs go red automatically, and the controller cannot override this.

The driver of the ARFF probably [1] placed more emphasis on the controller's clearance to cross than the lights telling him to stay put, in exactly the same way that the Russian air crew placed more emphasis on the controller's instruction to descend than their TCAS instruction to climb, not realising that they were maneuvering contrary to the thing that exists specifically to prevent these accidents.

EDIT: I am not assigning blame to the controller here. They are human, and humans make mistakes. That's why these systems exist. Having one person handle an airport the size of KLGA is an accident waiting to happen.

[1] Obviously this is unknown at this point, and is something the NTSB will investigate. The system could have been down for maintenance for example.