In the first film, Timon and Pumbaa bring Simba to the oasis after rescuing him from near death in the desert. They did not seem attached to any one place, but simply assumed home to be wherever they were. Simba grows to adulthood while living in the oasis and later encounters an adult Nala. Nala notes the place's beauty, but becomes confused as to why Simba never returned home. Simba is eventually convinced by Rafiki to return to the Pride Lands. After helping Simba and the Pridelanders defeat Scar and the hyenas, Timon and Pumbaa end up residing with Simba and Nala in the Pride Lands.
In the midquel, Rafiki unintentionally directs Timon toward the area, where, on the way, he meets and is escorted by Pumbaa. The two eventually arrive at the oasis while looking for their dream home. They name the oasis "Hakuna Matata" after the philosophy they both chose to live under upon arriving. The area apparently possesses everything Timon was searching for. He and Pumbaa claim it as their own territory, later raising Simba in the oasis. By the end of the film after Scar's death, Timon allows his former meerkat colony to reside at the oasis, thereby giving them a predator-free home.
Other animals are revealed to live here as well; for example, Kataka, Growler, and Zak from the "Hakuna Matata" books.
Simba, Timon and Pumbaa in the oasis
Pumbaa swinging on a vine in the oasis
Timon eating grubs in the oasis
Simba recovers on the border between the desert and the jungle
The trio walks across a log during "Hakuna Matata"
Simba mourns Mufasa on the edge of the jungle
Simba and Nala reunite in the jungle
A waterfall during "Can You Feel the Love Tonight"
Originally, Nala planned to stay in the oasis with Simba. She could've done this in the film had she not learned Simba had gone back.
The oasis is named after the song "Hakuna Matata," as Timon and Pumbaa chose to live there with "no worries."
According to the video game Kingdom Hearts II, the lionesses knew about the oasis, but Scar refused to let them leave to live there.
Mohatu could've visited this area to save his kingdom from the drought. According to The Lion King 1½, the oasis wasn't very far (for an adult lion).
Despite Timon and Pumbaa opting to live in Pride Lands along Simba after helping him in defeating Scar, the duo are still living in the jungle in the Timon and Pumbaa cartoon which takes place after The Lion King, but before its sequel.