One is that on a microscopic level, the diatomaceous earth particles are very sharp looking. These particles stick to an insect and get stuck between its exoskeleton joints. As the insect moves, it gets physically cut up.
The other explanation is that diatomaceous earth sticks to the insect and somehow causes them to dry out. I think this approach involves scratching the insects waxy layer which then allows precious moisture within the insect to get out. So their teeny tiny bug-innards turn into teeny tiny bug-innards-jerky.