Looking at the Pokemon who commonly use Teleport, you see things such as Clefable and Blissey. These are both fairly bulky Pokemon who have one thing in common -- they have a form of recovery activated by switching. Clefable has Wish, while Blissey has Natural Cure. While Wish doesn't activate by switching, it will heal its teammate by switching. If you have a low-health sweeper on your team and it won't be able to survive another hit, you may use Wish with Clefable. Clefable can take one more hit, so you stay in and use Teleport. Since it has negative priority, you sweeper won't take any damage before getting healed by Wish. Also note that Blissey can use Teleport to get rid of status due to Natural Cure. Next, onto Slowbro, Regenerator heals it when Slowbro switches. If you know that Slowbro can take another hit, you can use Teleport to pivot and your switch-in won't take damage (going back to my first point).
Teleport was also "buffed" this gen. While the move itself wasn't, the introduction of Heavy-Duty Boots is definitely one of the reasons for its popularity. While Clefable has built-in Heavy-Duty Boots (Magic Guard), Blissey and Slowbro, two common Teleport users, don't take damage because of Stealth Rock due to them.
Another reason why Teleport is popular is because its the only form of pivoting a Pokemon has. For example, Blissey is probably slow enough to be able to use Volt Switch / U-turn and deal a tiny bit of damage, but it doesn't learn either moves.
Hope this helps! :)