What is the tool you use to open the horses mouth and why is this necessary?
The tool in question is referred to as a full mouth speculum. All credible Equine Dentists will use a full mouth speculum to examine and to treat your horses teeth.
Treatment without a speculum is not recommended as the individual would try to control the horses mouth by pulling the tongue outside of the mouth to stop the horse biting down. While moving the tongue might be necessary in a controlled environment for an initial visual appraisal it is not acceptable for working in the horses mouth. If performed incorrectly, it can cause damage to the hyoid apparatus and this will be a problem for the horse. Also, it is only possible, if lucky, to access some of the front teeth this way and does not allow access to the molars.
You should not allow anybody to work in your horses mouth without a speculum. Almost all horses are happy to accept a speculum without any issues if done with care and patience.