A soul is an incorporeal, insubstantial "essence" that constitutes part of the definition of an individual human.
In distant prehistory, back in the mists of antiquity, those hominids who would one day be humans began to evolve what can best be described as a 'psychic organ'. While this organ had no physical component, it nonetheless was a real adaptation in neo-mankind's non-physical consciousness. This "mutation" bred true, and proved its survival value, and thus, was passed on to succeeding generations.
What this 'psychic organ' did, basically, was allow each individual hominid who possessed it to form a rudimentary connection to every other hominid who possessed it. In essence, it allowed our remotest ancestors to form a sort of primal telepathic network between themselves. This network wasn't powerful enough for most to be consciously aware of, so it became a sort of mass racial subconscious... but it was enough to allow those protohumans to form, accept, and internalize various social concepts, like the family, the tribe, the race, and eventually, the nation... and also more individual social concepts, like friendship, trust, companionship... and love.
There were other hominids around at this time not blessed with this random psychic mutation that jump started the social evolution of what would become humanity. The two groups rapidly grew apart, as those not gifted with 'souls' continued in their chaotic, egoistic, sociopathic, non-cooperative, violent and confrontational manner and humanity's direct ancestors, began to draw together into groups, teams, families, tribes, nations - social organizations aimed at mutual support and survival.
Several dynamics evolved between the two distinct groups. When the social protohumans with 'souls' could be caught alone, they made fine and easy prey, because their new 'empathy sense' weakened their naturally violent and viciously individualistic natures. They were naturally inclined to 'trust', to try to help someone they saw in need, and to behave in other completely irrational and utterly insane ways...which would allow a cunning soul-free hominid to trap them easily. In general, this dynamic worked to create, enforce, and increase hostility and distrust between the soulful and soul-free populations, and to force the newly socialized protohumans to draw together even more tightly, from outside predator pressure. The protohumans were driven into migrations by such predator pressure into new geographic regions free of the attacking soul-free creatures, and the intense competition between the groups would impelled the protohumans to develop new innovations in defense - new technologies, new organization techniques, better methods of communication, improved weapons - which had a synergistic inter-relationship with the fact that the protohumans, being already enmeshed in a psychic, racial subconscious telepathy web, already had a far better capacity for cooperating in the invention and implementation of such innovations.
The end result was that human evolution - social, intellectual, technological, and even genetic - tended to leap ahead of that of their soul-free fellows, and that humans themselves became far more inclined to an orderly approach to their day to day life, with higher and higher levels of social organization arising from each succeeding one. Increasing Order successively rose from previous chaos, in a reversal of the normal entropic process. While 'demons', the soul-free populations, continued to be mostly chaotic and unorganized, fighting amongst themselves for dominance and being able to, at best, organize themselves into mutually beneficial cooperative unions for a very short term. Those individual 'demons' smart enough to think up innovative and effective advancements in any area kept them to themselves, and thus, the demonic race as a whole and as a collective did not benefit from the genius of its individual members.
Eventually, humanity, with its strange, unforeseen 'psychic organ' mutation, came to utterly dominate the Earth.
Basically the 'soul' is a non-physical component of individual human identity evolved by our distant, protohuman ancestors that through a subconscious telepathic network link with every other human, gives us all a rudimentary social consciousness - a concept that we are not alone, that we belong to something greater than ourselves, that the group can be as important, or more important, than the individual, and most important of all, that another individuals, or every other individual, is as important, and valid, as we ourselves are.
Demons don't have this, and given that, from their viewpoint, it causes humans to behave in frankly irrational manners that make them extremely vulnerable to exploitation and depredation by those without such an inherent psychic flaw and weakness, they don't want it. The enormous advantages given to humanity by our natural aptitude for organization and team work are things that demons themselves can only dimly grasp and not really comprehend; they understand that somehow humans have come to dominate the Earth, but they aren't sure how.
When a human being is converted to vampirism, their body transforms from normal flesh and blood into a near-exact duplicate composed of necromantic energy, but the soul, being an entirely immaterial, psychic organ, is not duplicated. Thus, the human self still existent within the transformed body finds itself without that vital aspect of its individual consciousness that it has always had, and as such, no longer has any instinctive feeling of sympathy, empathy, or socialization. There is no longer a capacity for true friendship or love in the human who now finds him or herself in a vampiric body; they are cut off from the telepathic racial subconsciousness. That sense of belonging to something greater is gone.
The soul is a product of evolution that gives humans a huge psychic advantage over demons. It's a natural part of their intrinsic humanity.