A tailwind means that the starter can't spin up the engine as quickly as is normally scheduled.
This means that the EEC commands the FCU to pump more fuel into the engine to maintain the scheduled engine acceleration.
More fuel = higher EGT.
Less RPM = less compression which means less expansion of the gasses thus higher EGT. Similar to why a TC piston engine has higher EGT than a TN/NA piston engine due to the lower compression ratio.
PV=nRT
https://en.wikipedia.org/wiki/PV%3DnRT