Avatar: The Last Airbender, not to be confused with James Cameron's Avatar sequels, focuses on four different nations of people who are able to control the elements of Fire, Air, Earth, and Water respectively. Only the Avatar, an ancient being who has been reincarnated through all the ages of their world, can harness all four elements.
This ultimate being is responsible for keeping harmony among the four nations of benders, and so when Aang becomes encased in solid ice for over a hundred years, it presents the Fire Nation with the perfect opportunity to tip the scales in their favor. They wage war on the unsuspecting peoples in surrounding lands, in order to solidify themselves as the most dominant and powerful race, and they start this conquest by committing a terrible act of genocide as they entirely eradicate the Air Nomads from existence. But why?
The Avatar cycle always follows the same order, so when the last Avatar, the fire-bender Roku, passes away, they know that the Avatar’s next life will be born into the Air Nation. The Avatar is the most powerful being in their world, and the only one who would be capable of single-handedly defying the Fire-Nation's bid for control. So there is plenty of motivation for the Fire Nation to take an interest in capturing Aang, and stopping him in his tracks, but could this really justify the murder of his entire race of people?
One possibility as to why they might’ve killed the Air Nomads is that the Fire Nation wanted to capture Aang whilst he was still a child, and keep him locked away somewhere, so that he could never reach his full potential, and the Air Nomads refused to reveal his location. Even though the Avatar is born with the ability to harness all of the elements innately within them, they are a skill that takes much patience, dedication and love to master. If the Fire Nation had locked Aang away before he was ever able to learn how to control his abilities, they would have had free rein of their desire to wipe out all others who stood in their way.
In fact, once they had wiped out the Air Nation, they began systematically eradicating the water benders too, which is why Katara had to journey so far to find the most powerful water benders in the Northern Water tribe. She was the only water bender left in her tribe, and even then, only because her mother had hidden her and died to protect her. It, therefore, stands to reason that the Fire Nation would have slaughtered the Air Nomads for not giving the Avatar up and revealing his whereabouts so that they could take him into their custody and prevent him from ever becoming a threat.
Alternatively, it could be that the Fire Nation wanted to kill Aang, and the next two descendants who came after him. After Aang’s death, the cycle would then pass into a water bender, and if they managed to track and kill this child too, then it would pass into an earth bender, and finally, back into the Fire Nation once more. With Roku at their helm, the Fire Nation was becoming impossibly strong, and could very quickly have entered into a state of total domination over all other elements. Luckily, the previous person in the cycle, Avatar Kyoshi, taught Roku that his Avatar duty was to protect all people, and to keep the balance of peace between the elements, and he prevented many terrible disasters from taking place. But if the Fine Nation could force the Avatar cycle back into a child as a fire bender, and raise that child with the mindset of wrath and destruction, they would have the ultimate unstoppable weapon at their disposal.
The third, almost more chilling reason could be to do with the show's unusually deep lore surrounding the Avatar state. In wanting to kill Ang, the fire nation could have been harboring a desire to end the Avatar state completely. When in the Avatar state, the bender is at their most powerful, because they are able to tap into all of the knowledge, experience, and strength of their past lives combined. But this is also when they are the most vulnerable because the Avatar state makes them less tethered to their earthly bodies.
If an Avatar is killed in the Avatar state, it is believed that the cycle will be broken, and that the Avatar will never again be born. This would eliminate all possible threats to the Fire Nation forever, and would spell disaster for all peoples of their world. As a proud nation, that considers itself superior to other benders because they are the only ones who can create their own element, the mere existence of an Avatar, a being who will always be more powerful than their forces put together, is an insult. Therefore, in killing the Avatar permanently, the Fire Nation would prove once and for all that they are the supreme race. And that’s a title that is worth killing the Air Nomads for.