Why does freshly slaughtered chicken needs to be cooled down before cooking?
The consensus from sites I have seen that appear fairly credible is that you need a resting period to avoid toughness in the final product.  I speculate this is to allow rigor mortis to come and go.

According to Girl's Guide to Butter's article Chicken Butchering 101:


  And now the final step in the butchering process – put them in a
  larger tub filled with cold water (ours comes straight from the ground
  and is very cold), and let them sit and relax till the evening (if
  you’ve butchered in the morning) or overnight ...
  
  Note that it is important to let your chickens rest and relax in this
  fashion before cooking them, because, if you cook them right away (not
  that most of you, unlike me, would want to), they are likely to be
  tough.


Similarly, The Self Sufficient Home Acre's How to Butcher a Chicken article says:


  [P]ut it in the refrigerator for 24 hours before cooking. If the meat
  is not allowed to relax, it will be tough. If you plan to freeze the
  chicken, you can leave it in the refrigerator overnight or you can
  freeze it right away. Just be sure to pull the frozen bird out and let
  it thaw completely before cooking it. I have had good luck with this
  method.