Edit: Surely we can come up with a more approachable explanation for less technical folks, though? Here's an attempt:
"Contact tracing respects your privacy and does not send your location to the cloud.
Instead, your phone makes up a new random name every 15 minutes and broadcasts it to nearby phones. It remembers the last two weeks of names it used, as well as the last two weeks of names it heard from other phones.
When someone catches COVID-19, they register it in the app. Their phone then uploads the last two weeks' worth of names it used to the cloud, where other phones can download the data. The names aren't connected to their identity, all they represent is someone who caught COVID-19.
If your phone finds a match between a name it has recently heard and the online database, it sends you a notification. After 2 weeks the data is erased, so you are only notified if you were near an infected person in the past 2 weeks.
Since the random names change every 15 minutes, nobody can track you or know that you are the same person as last time they saw your phone. The data is only stored locally, so after it is deleted two weeks later, there is no way to go back and recover it."
How's that?
(Edited because without the intro sentence it sounded like I was trying to imply the parent didn't get it; that wasn't my intent.