All Ps are not Hs. This is because all Ps are Ts, and all Ts are not Cs. (Contrapositive: all Cs are not Ts.)

The conclusion is about not Hs, but none of the premises mention Hs. We need to connect what the premises tell us about Ps (as a sufficient condition) to not Hs (as a necessary condition).

We can infer from the premises that all Ps are not Cs. So, suppose that all not Cs are not Hs. (Contrapositive: all Hs are Cs.) We could then infer that all Ps are not Hs. And the author’s conclusion would be guaranteed.

Note that assuming all Ts are not Hs would also guarantee the author’s conclusion. But it’s more common on the LSAT for the right answer to use all the premise connections available from the stimulus.