What you really mean is that a photon " is both wave like and particle like'. The most obvious physical property is what you choose to detect it.....Different circumstances tease out one property or the other.
You simply have to read accepted experimental evidence which reveals current understanding.
None is better than the double slit experiment [wave particle duality] and the photoelectric [particle] effect explanations. Wikipedia has the necessary insights regarding each.
These are not necessarily intuitive and arrived at via some suddenly "EUREKA" moment...but are observational, empirical results. It's the way stuff works and that's why quantum mechanic models are the way they are...to mimic what is observed experimentally.