Where does the word mesmerise come from?

The word mesmerize is named after Franz or Friedrich Anton Mesmer 1734-1815.
mes·mer·ize  (mzm-rz, ms-)

tr.v. mes·mer·ized, mes·mer·iz·ing, mes·mer·iz·es

1. To spellbind; enthrall: "He could mesmerize an audience by the sheer force of his presence" (Justin Kaplan).

2. To hypnotize.

The connection to hypnosis is fascinating -  Mesmer had an exotic and unconvincing theory about animal spirits/animal magnetism being the key to medical treatment. 

In 1774 during a magnetic treatment with a female patient, Mesmer felt that he perceived a fluid flowing through the woman’s body whose flow was affected by his own will.He eventually named this fluid and its manipulation “Animal Magnetism” and developed an elaborate theory regarding its affect on health. source

Animal Magnetism is not a recognised branch use of hypnosis for theraputic purposes was pioneering.