The "Lions and Tigers and Bears", as they are first introduced by Ummon of the TechnoCore, are a group of mysterious and extremely powerful entities that reside in the medium known as The Void Which Binds.
The phrase "Lions, tigers and bears" is a direct reference to The Wizard of Oz, a quote from the main character Dorothy and is a reference to an unknown feared predator which may or may not actually exist.
SPOILER ALERT! The section(s) below are MAJOR SPOILERS.
The origin of the Lions and Tigers and Bears is never fully explained; it is mentioned that they are made up of both organic and synthetic species, differing in their backgrounds but united through empathy. It is possible that the human Ultimate Intelligence, which is supposedly derived from the consciousnesses of all human beings, may shed light on how the Lions and Tigers and Bears came about.
As the virtual entities of the AI (Artificial Intelligence) TechnoCore evolved, they decided to relocate from the human computer network which stored their personalities (the Internet), thus ceasing their dependence on humans. Their idea was that The Void Which Binds, a quantum medium that they had previously used to supplement humanity with technological innovations, could host them as abstract waves of probabilities, hence sparing them the need of dwelling in any sort of physical terminal. No longer having any need for humanity, the TechnoCore planned to orchestrate an "accidental" catastrophe which would destroy Earth after their retreat.
During the AIs' forays into the Void Which Binds, they encountered other intelligences - "Lions and Tigers and Bears", as they later described them. These beings were immensely powerful sentient entities whose consciousnesses resided in the Void. Their gargantuan strength comes, it is explained, from quasars and blasting cores of galaxies, as well as competence in manipulating space and time through the Void. They use love and empathy to merge with each other and with the universe, as the ability to experience such concepts enable one to access the Void and constitute mental connection to every being and location at all places and all times. They are thus united in true symbiosis in the Void, and therefore stand in bold contrast to the AI which evolved as parasites, and are hence incapable of feeling empathy. The Lions and Tigers and Bears did not want to host creatures who lack empathy, since they cannot participate in true symbiosis.
This encounter, unpleasant for the AI, ended in them fleeing out of the Void, secretly stationing themselves at the Megasphere, the intermediate layer of information through which their technology functions. Although the TechnoCore had already commenced the destruction of Earth by dropping a black hole into its core, the AIs desperately tried to halt the catastrophe and engineered a way of negating the black hole.
The Lions and Tigers and Bears, however, had other plans for Earth. Using their ability to manipulate the Void Which Binds, they freecast Earth to one of the Magellanic Clouds. Oblivious to what had transpired, humanity thought Earth had been swallowed by the black hole; the TechnoCore perpetuated this myth, secretly terrified by the power of the beings that had "stolen" Earth.
Even though the AI Technocore was unwelcome, the Lions and Tigers and Bears, as extremely empathic creatures, gave humanity a chance to join them. In order to determine whether or not humans are capable of the empathy necessary to fit into true symbiosis with other sentient species and with nature, they used Earth to conduct an experiment. They inhabited it with genetically-developed artificial humans, and instructed them to reconstruct an ecologically healthy Earth, which is able to host both human and other life forms peacefully; in doing so, they would determine whether humanity would have been able to develop into the kind of species they wanted, had their planet not been destroyed and their destiny altered. The success of the project resulted in the Lions and Tigers and Bears' sending cybrid John Keats to find a human female to reproduce with. The offspring of this union produced Aenea, which combines the human quality of love and empathy and the AI quality of accessing the Void Which Binds. Aenea was a "virus" planted by the Lions and Tigers and Bears inside humanity in order to "infect" them with the ability to access the Void Which Binds, so that they can join the Lions and Tigers and Bears in true symbiosis.
Nature and Motivation
The identity of the Lions and Tigers and Bears is never made clear throughout the Cantos. It is only mentioned that they are all sentient species capable of empathy, part of them organic, others synthetic, and that they have inhabited the known universe and the Void for billions of years. It is unknown whether they have bodies, and if they do, where they reside. Their purpose seems to be making every spot in the universe inhabitable by creating biospheres around stars, star systems and eventually galaxies. Biospheres are bio-technological environments which are balanced with both their location (outer space) and their dwellers, with the dwellers being evolved in such a way as to fit into their biosphere.
The Ousters are in a pact with the Lions and Tigers and Bears. Their evolution is influenced by the nanotech which is adjoined to their genes and causes them to adjust themselves to their environment. This kind of lifestyle is what the Lions and Tigers and Bears require from humanity, and this is what Aenea is supposed to eventually deliver to humanity.