More Human Than Human
Human motivations have been chronicled, studied, categorized and explained for hundreds of years and as a result, there is a wealth of reading material to source possible motivations for hating another culture and wanting it dead.
Our will to survive is legendary, even amongst our own kind. I'm jesting, but a common theme in literature is the indomitable spirit of a Human trying to survive. Whether one's life is in actual or perceived danger doesn't diminish the response. Ultimately, this means that in your world, humans must have believed at one time that the other races/creatures directly, or indirectly, threatened their own race's survival.
Competition for Resources
The scarcity of valuable resources drives competition and this applies to all levels of humanity. If you have your world organized into nations of people who identify with each other geographically, then they may have trade relations, a marketplace for resources, and an economic dependency on each other. Put humans in charge of one or more of the government bodies that have force/violence capabilities. If one of your nations was ruled by monarchy/dictatorship and had sufficient power, you could start a global, genocidal war based on one human ego.
The Human Queen, Veruca Salt, has always wanted to ride a dragon. Since the Last War of Good V. Evil, dragons have been bred down from intelligent, arcane magic-wielding people-eaters, to cherished, lifelong companions by the Elves. The dragons are paired with Elves during a sacred tradition of linked, simultaneous birth, named The Clarion. The Clarion is eight millenia in practice, which is old even for elves. The ritual synchronizes the hatching of the dragon from its egg and the birth of the elf from its seed pod through the careful application of mystic energy the Elves refer to as Wu Wei. Inexplicably, only elves feel and can manipulate Wu Wei, which is done so almost exclusively by priests of the elves most holy order, The New Dawn. Humans will never train dragons, because the only dragons left are those the elves pair with. Sooner or later, Veruca Salt starts a war to get one. This begins the initial, hateful divide.
I think the 2014 American film Maleficent is an excellent take on this concept.
Truthfully, once the initial spark has set things on fire, humans can be notoriously difficult to talk down from war. Religion can be started by a single, charismatic lunatic, or a narcissist. Cultures change with exposure to meta-stimulus over a few generations. Contagions make entire species a threat to the common health. One race's technology/magic could run awry and cause a catastrophe that kills a disparate amount of another race. Maybe they just don't like each other and that's enough.