Is it safe to put food in the fridge shortly after cooking?

My wife insists that just cooked food must be left on the counter to
cool down before it is put in the fridge, otherwise it is prone to
spoil. She says this is a well known fact amongst all fine people who
cook. In fact, she claims, "any culinary school will tell you this,
it's common fact and just plain common sense."

Personally, this completely contradicts what I consider to be common
sense. Heck - I was just trying to make room on the counter!

This topic has come up a few times now and I'm hoping to get someone to
back me up, or set me straight.

-Colman