Why does a phase shift in a light pulse imply a non-commutative structure of space (which implies gravity has a quantum structure)?

Quantum mechanics, in its most basic form, relies on the fact that position and momentum don't commute, $$ [x_i,p_j] = ihbardelta_{ij} $$ (where $x_i$ and $p_j$ are the $i$th and $j$th components of position and momentum), which in essence encodes the de Broglie relation and the Heisenberg uncertainty principle, and the fact that position and momentum are incompatible observables.

Certain theories of quantum gravity include modifications to spacetime that make it 'noncommutative', which means that there are nonzero commutators $[x_i,x_j]$ and $[p_i,p_j]$ (as well as modifications to $[x_i,p_j]$), so that the different components of position and momentum are no longer compatible observables.

This is a fairly drastic change to quantum theory, and it is easy to see how such modified commutation relations would impact the predictions. The paper in question (Nucl. Phys. B 924 578-587 (2017), arXiv:1710.03920) works through the impacts of such a modification and shows that it is (at least in principle) observable using optomechanical observables.

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Can anyone explain why a phase shift of the light pulse implies a non-commutative space structure?

It doesn't. It provides an experimental test that can reveal configurations which are incompatible with standard quantum mechanics and which could be explained with a specific modification thereof. (Assuming, of course, that experiments were indeed to coincide with the authors' predictions and fall outside the range explainable with standard QM, which is not the case yet.)

Such an experiment would be the first piece of evidence towards establishing the theoretical framework used by the authors, but it would require a mountain of additional evidence to "imply" a non-commutative space structure.