It is the Avatar's duty to master the four elements and protect the balance of the world. At the end of their life, the Avatar is reborn in another nation and repeats the process all over again. However, reincarnation and responsibility aren't the only constants in the Avatar's life. Both Katara and Roku have confirmed that an animal guide is something that every Avatar receives -- each Avatar has an animal companion to help guide them. This animal seems to be spiritually bonded to the Avatar in both life and death.
In addition to usually being an adorable mascot for the Avatar's team, this animal companion is also extremely loyal and helpful to the Avatar. The Avatar Spirit, Raava, grants the Avatar many privileges and benefits, such as vast power and experience. So why is it important for the Avatar to have an animal companion?
Tradition
Tradition and rituals are very important in the world of Avatar. Certain practices are kept alive for hundreds of years simply to honor the past. Each Avatar might have an animal companion simply because the first Avatar had one. Wan, the first Avatar, went on his journey with a cat-deer named Mula. After Wan saved Mula from a group of hunters, he became Wan's close friend and mount. Mula traveled the world with Wan and aided him in his Avatar duties. Raava could be ensuring that each Avatar has an animal companion to honor Wan and Mula's relationship.
Transportation
The Avatar is responsible for protecting the entire world, which means they need to travel large distances quickly. All of the animal companions that audiences see in the series are large enough to ride. During the Hundred Year War, Aang traveled across the four nations on his sky bison Appa. Roku used his dragon, Fang, to quickly respond to emergencies all over the world. The Avatar's mount allows them to journey across the four nations much more conveniently than if they had to travel on foot.
Protectors
The Avatar has an incredibly dangerous job that draws the attention of many villains. They are often the victims of kidnappings and assassination attempts when they haven't yet mastered the four elements. Korra was almost kidnapped by the Red Lotus as a child before she learned to defend herself. Raava could be choosing animal companions that are capable of protecting the Avatar from such attacks. Both Appa and Naga have fought alongside the Avatar and displayed a fierce loyalty to them.
Teacher of the Elements
Many of the animal companions of the Avatar are able to bend the elements themselves or else represent them in some way. These creatures taught humans how to use their bending powers. Dragons and sky bison were the first benders of the fire and air elements respectively. Fang and Appa could have served a similar role for Roku and Aang. The animals may have acted as personal teachers to the young Avatars, showing them how to use their native elements. Even Korra's polar bear dog Naga would be familiar with the icy terrain of the South Pole and could aid Korra in her waterbending training.
A Life-Long Friend
The Avatar's animal companion is much more than just a form of transportation or a bending teacher. The companion is also one of the Avatar's closest and earliest friends. Aang and Korra met their animals at a young age and soon became inseparable. It's possible that Raava wanted each Avatar to have a friend that they could rely on for their entire lives. The Avatar spirit could be linking the spirits of the animals to the Avatar so that they could never be separated. Fang died alongside Avatar Roku and continued to be his animal guide in the afterlife. Appa more than likely died at the same time Aang did since they were spiritually connected. It's comforting to know that no matter what happens, the Avatar will always have a friend by their side, both in life and in death.
