Why does coNP⊆NP∖P imply that the polynomial hierarchy collapses?
I believe the claim is more commonly written as $coNP \subseteq NP/poly$, but that of course does not immediately answer your question.  

The fact that this implies that the polynomial hierarchy collapses is shown in Theorem 2 here:

Chee-Keng Yap. Some consequences of non-uniform conditions on uniform
classes. Theoretical Computer Science, 26:287–300, 1983. doi:10.1016/
0304-3975(83 ) 90020-8

If I understand correctly, $coNP \subseteq NP/poly$ is shown to be equivalent to $coNP/poly = NP/poly$ (unnamed corollary on page 292), and this would then imply $\Pi_3 = \Sigma_3$ (Theorem 2), implying collapse of the further hierarchy. For an actual proof, if you have access to the paper, I'll refer you there, I am not familiar enough with it to attempt an explanation now.

On an unrelated note, I believe the paper you are citing to be incorrect (but that would probably warrant a separate question?)