What does "thanks x-institute for hospitality" mean in an acknowledgement?
Usually it means that, during at least part of the preparation of the paper, the author was on an extended visit to Institute X, and X provided something that was helpful.  It could be as little as an office or a library card, or as much as free housing, use of expensive equipment, travel support, etc.  However, this phrase wouldn't normally be used if the author was actually a paid employee of X during that time, even if temporarily.