This is esoteric knowledge, only for people interested in how an organization can offer its own ITIL training.
Usually, if an organization wants to hold an ITIL training class, they will get on-site trainers to come train the organization. This page does not address how to get an on-site trainer. This page describes how an organization can actually offer ITIL training itself.
This is not about an organization becoming ITIL "certified." Organizations cannot become certified in ITIL--the closest thing to an organizational ITIL certification is passing an ISO 20000 review.
Someone (I'm not sure who, but perhaps the ITIL Qualification Board) sets a very small list of ITIL "Examination Institutes." One of these Examination Institutes is a company called "Loyalist Certification Services." Loyalist is one of the North American Examination Institutes.
Loyalist Certification Services in turn creates "Accredited Training Organizations (ATOs)," "Accredited Trainers (ATs)," and approves "Accredited Training Associates (ATAs)" that are attached to ATOs. This accreditation includes a list of classes that each organization can offer.
Loyalist also certifies ITIL training materials--any slides used in an accredited training class are reviewed and approved by an Examination Institute.