Counterfactual computation

Being trained primarily in biology, quantum physics is not my strong point. I find it fascinating, I enjoy reading about it and I wish at some level there was an intuitive way for a non-specialist to understand it. The problem is that even practitioners in the field describe it as counter-intuitive.
Today I’m trying to get my head around a Nature article, Counterfactual quantum computation through quantum interrogation. The essence of it, summarised here, seems to be that due to the probabilistic nature of photons, you can get them to cause effects without them actually being there. In this case, obtaining a result from a quantum computer without running its program.
No, I don’t get it either, but it intrigues me. Any physicists out there care to profer a simple explanation?