Why is the word hospital in the word hospitality?

Answer 1

Because hospitals take care of you, that's what hospitality is.

I'm sure there's a better answer using the latin root word(s), but that's the gist.

Answer

The word "hospital" comes from the Latin "hospes" which refers to either a visitor or the host who receives the visitor. From "hospes" came the Latin "hospitalia", an apartment for strangers or guests, and the medieval Latin "hospitale" and the Old French "hospital." It crossed the Channel in the 14th century and in England began a shift in the 15th century to mean a home for the elderly or infirm or a home for the down-and-out.

"Hospital" only took on its modern meaning as "an institution where sick or injured are given medical or surgical care" in the 16th century. Other terms related to hospital include hospice, hospitality, hospitable, host, hostel and hotel. The Hotel-Dieu, a name often given to a hospital in France during the Middle Ages, is the hotel (of) God.

source: http://www.medterms.com/script/main/art.asp?articlekey=8390