The duality of electrons and light, i.e. existence both as particles and waves, is a breakthrough Physics achievement through which we can explain practical observations.
de Broglie suggested that electrons exist as both particles and waves, while the photoelectric effect (by Einstein) postulated that light can also exist as photons or quantas (in addition to its wave-like nature). A single nature, either as particle or wave, for electrons or light could not explain all the practical observations. For example, photoelectric effect and black body radiations can only be explained by the particle nature of electrons. Yet it could not account for the diffraction and polarization. That can only be explained by the wave nature of the electrons. The dual nature idea originated from the fact that electrons accelerate as they move around nucleus and hence will lose energy and fall into it, thereby neutralizing the protons. Since this does not happen, it must be because electrons behave like standing waves. Think about the dual nature as an illustration of personal-professional life of a person. A teacher is not considered a father in a class full of students, while the same teacher will not abe considered a teacher at his home by his children (to them he would be their father). Thus, the same person exhibits two different identities, depending on the situation.
Thus, the idea of duality of electrons and light help us in explaining the practical observations.
Hope this helps.