It's actually quite simple, the aerobridge deflects airconditioned air (either aerobridge or cabin) which then escapes around the seal to the aircraft directing air close to the aspirated TAT probe. The TAT reading is correct, the air is not at the ATIS temp. On a hot day TAT under-reads, and cold morning TAT over-reads. This alters the FMC computed figure, and disagreement with the EFB. After pushback, the TAT again measures the ATIS OAT and the two figures resolve.
The really surprising thing is the lack of acknowledgment that the anomaly exists, and the lack of further questioning as to why. As I said, the answer is is on the upper DU, with a simple cross check of TAT vs ATIS to reason the source of the discrepancy.
Acceptance of discrepancies without question can lead to these types of incident.