It's been discussed before on Pprune - maybe a year ago.
Essentially it's this: Rotors are massive forgings, and heat up & expand slowly. Casings are light by comparison, and heat up & expand quickly.
During takeoff, the differential in thermal growth produces a temporary increase in rotor-to-stator clearances, which are leakage paths for the airflow in the engine. More leakage => lower engine efficiency => higher EGT.
The static runup tends to help get the rotor warmed up, and clearances close to normal, before the takeoff begins.
That said, the most modern engines often have some active clearance control scheme to help mitigate the above processes; but the warmup technique will still probably help some.