What does this A320 FCTM quote about cold weather and non-precision approaches mean?
I am not sure which part you don't understand, but here is my attempt to explain in more laymen terms.



Short version: When it is too cold, a certain mode for flying non-precision approaches using the Airbus flight computer cannot be used. This is because the altimeter reading is inaccurate when it is too cold, but the computer is not designed to compensate that.



Long version: For all non-precision approaches (i.e. approaches where ground equipment is not providing glideslope signal), there is a minimum Outside Air Temperature. 

Since the reading of the barometric altimeter is affected by temperature, when it is too cold outside, the reading can no longer be considered accurate enough for the safety of flight. Corrections should be made (using a mechanical flight computer / lookup table / other tools) to convert the indicated altitude on the instrument to the reading it would have indicated under normal temperature.

Since it is not authorized to make these corrections in the autopilot computer, it is not possible to use the "FINAL APP" mode to fly the approach when the outside air temperature is below this minimum.