Imagine you want use Google Home to manage your home (questionable decision, but bare with me for a moment). Whenever you get near your house it should turn on your lights and starts up the heating so it’s nice and toasty when you get home. In order to do this you need to constantly send your GPS data to Google so they can track when you get near your home. Even if your location data is slightly obfuscated (e.g. by rounding to nearest mile or some hexagonal grid) they still get a rough estimate of your location. What this paper uses is a “thing” called SNARK. You can think of it (for this example) like a function that computes if you are near your house or not. You then execute that SNARK on your local device with your current exact GPS coordinates. The result of that operation is a signed result that is the proof if you are near your house or not, without actually telling where exactly you are. This proof can then be sent to Google without much fear of giving them any data they don’t need. The privacy aspect is especially interesting whenever you are not nearby: it will just tell Google that you are not at home, they have absolutely no idea if you are at work, or in Australia, or on the Moon or wherever else. I think the main thing they did in this paper is to define some operations on SNARKs that lets you compute proximity efficiently.
I tried to answer another question which should also work as a summary: https://lemmy.world/comment/13409438