When two high energy photons are smashed together they create a matter-antimatter pair. The photons do not interact with the higgs field, but when they convert into matter-antimatter pair, they start interacting with it.
I am not understanding how something not interacting with it starts interacting with the Higgs field.
You are asking the reader to accept a bad picture that could never make sense.
In your thought experiment, the two photons convert into several varieties of virtual fermion pairs which then recombine to form a Higgs. (Actually, in the lab, LHC/CERN, the time reversed process happens, the Higgs decays to two photons, instead, quite rarely.)
This is technical language reminding QFT users how to calculate the frequency/likelihood of that event: there is no "start" of the interaction. The interaction is a feature of the Higgs particle and the photons and is there in the list of answers, "right from the start": the Higgs simply splits into two photons in an instant. The intermediation of that coupling through a fermion-antifermion pair is a symbolic memo to calculators on how to estimate the coupling from the SM Lagrangian, the starting point of their estimates. You appear to be stretching a shared metaphor for a calculation to something beyond a helpful fantasy.
