How do cold conditions help organ transplants?
This paper actually goes into the whole history of organ transplants.

In short it seems to have the following effects:


preservation - usually with a specific solution to help.
slows down extracorporeal ischaemic damage
slows down hypoxic damage 
slows down the metabolism (energy consumption) and thus the need for oxygen that blood provides.


Remember that the cooling doesn't allow the organ to live indefinitely, and may only work for a few short hours depending on the organ. 

I highly recommend reading the paper if you'd like to know more about organ transplant. They went into great detail about the past and present and the abstract also says they discuss new techniques about to be don't in clinical trials.