The group were an international terrorist organization that made its business by causing chaos on among the warring nations of their world. They saw a threat to their existence in the pacifistic, logical teachings of Surak as his ways would have meant that no nation would attempt to undermine its competitors frontiers, economy or leaders.
The unified coalition of nations asked Surak to meet with the Yhri on their behalf in order to discuss a lasting peace. Surak complied and met with the Yhri who welcomed him graciously on the camera's after which they captured him and ultimately killed him which became one of their most infamous acts in Vulcan history.
Rather then face attacks on themselves, the Yhri were confused to discover that the other nations sent emissaries that negotiated for peace. Many of these emissaries were killed but a year later, the Yhri would abandon their ways as the many nations of the world changed their ways to embrace the peaceful teachings that the departed Surak had taught.
The Yhri's motives were never fully explained with many of the groups members talking at length about Surak before going into self-imposed exile or ending their lives. Members of the organization stated that no matter what they did, they encountered something that was associated with Surak that reminded them of their horrific past yet still forgave them. (TOS novel: Spock's World)
- This somewhat contradicts ENT which, in its fourth season, said that Surak was killed by radiation poisoning during the Sundering due to conflict with the army that "marched under the Raptor's wings". However, its entirely possible that the Yhri were composed of the group and were responsible for detonating the nuclear weapon.