Samkha:
I believe there is a misunderstanding of the fundamentals of process heat transfer going on – or perhaps a typo error.
The thermal energy in a hot fluid is transferred to a colder fluid. It can never be transferred from cold to hot. It is the hotter fluid that has the higher enthalpy and the enthalpy is the driving force that transfers the thermal energy. The process is analogous to electrical flow: coulombs flow from a high voltage reference to a lower one. So do Btus. It can never be the other way around. You are right in questioning the statement “energy transfer by hot fluid to Cold one”.
The temperature approach (the temperature difference between the incoming warm fluid and the outgoing cold fluid) to which you refer is kept as close as possible in a heat exchanger in order to maximize the amount of heat transfer. That explains the reason for designing for a very close temperature approach in cryogenic processes. Of course, the more heat that is transferred, the larger the required heat transfer area. This is accordance with the basic equation: Q = U A (LMTD). Therefore, the main factor that comes into play in selecting a close approach is economics - as Pilesar has explained. The cost of refrigeration in a cryogenic process is the driving force that decides to what extent you can justify large heat transfer areas in order to maximize the heat transfer.