Nobody knows at the moment. Theories exist, however. Here's mine. Note this:
They are known, however, to have feet, as revealed in Pokémon Mystery Dungeon: Explorers of Time and Explorers of Darkness and Pokémon Mystery Dungeon: Explorers of Sky. ~Bulbapedia
So Dugtrio has some type of feet below the ground. One can then say that they possibly have some type of claw to aid in its digging. Those claws are what allow it to use Cut, Slash, Night Slash, and Scratch.