Hello,
The previous users are correct, it's the running that causes the grass to wear away. That being said, what we've found is that turning the game off, back on, and then running over that same patch of grass makes it wear away even more quickly. For a lot of players, they might play one or two rounds of Animal Crossing a day and never see any paths form. Players who switch characters often or play multiple rounds a day may notice their grass disappearing much more quickly.
The good news is that the grass will grow back, albeit slowly. To expedite this process, players will want to plant trees and flowers around their town. Also, as other users have suggested, players will also want to avoid running on the bare spots, which will prevent regrowth. We understand that this can be tricky in commonly traveled areas (near Tom Nook's, ramps, or the player's house for example).
As always, you can contact us at 1-800-255-3700 between the hours of 6am and 7pm PST any day of the week to discuss the situation with one of our representatives. Thank you.