First and foremost, you need vision to be able to maximize the range of your tempest. Oracle's revelation (like grubby does in the link) or an observer will provide you with this. Tempest are pretty expensive, take a lot of supply and are pretty late tech. You do not want them to be alone. You have to view them as flying siege tank. The tempest are pretty fast for how big and strong they are, but their slow rate of fire and low damage vs non-massive air will make them weak vs smaller units like viking and corruptors.
So, like I said before, use them like siege tank, literally. You can have 4-5 hover over your army, get vision of the location you want to attack with an obs or revelation and start shooting your target. If your opponent wants to engage the tempest, he will have to engage your army AND the tempest. Just like with Terrans and siege tanks, as long as your tempests are not out of position, winning the engagement will be hard for your opponent.
Situations they will help you in:
Terran is going mech, you can shoot the tanks/Thor from afar.
Killing that really annoying PF and forcing the engagement out of the range of said PF
Against siege lines, no longer do we have to send Zealots from all direction and hope for an engagement that will not be too bad for us, we can force it now.
Broodlords army, you can shoot the BL from afar
Snipe a hatchery for a favorable position
Snipe ultras or Swarm Hosts
The main weakness of the Tempest is their lack of splash damage, so your supporting army should have some colossi or Templars since a protoss deathball without splash isn't that hard to kill.
These are the situations I can think of, as I see more uses of them in tournaments, more situations might come where they are useful (pretty rare to see them because of how far in the tech three they are and they don't have splash damage).
EDIT: People asked for a game:
European launch event
put the video at 1h13 mins for the Grubby vs stephano game