Non-mathematical? I'll give it a shot.
When the object beam and reference beam intersect on the plate, they modify the optical properties of the recording medium that is on the plate: either they cause the optical absorption to change, or they cause the refractive index to change, according to the intensity of the light at each point. Interference between the object and reference beams produces a pattern of light and less-light fringes on the plate, so the recording medium acquires a corresponding pattern of dark/light fringes or high/low refractive index fringe.
Denis Gabor earned his Nobel prize for realizing that the fringe patterns thus recorded amount to diffraction gratings that, upon illumination with either one of the original two beams, will reconstruct the other beam.
As for why the recorded fringes diffract light precisely that way, it's much easier to explain in mathematical terms, but it can also be explained via Huygens principle. If you look at Figure 2 on this page, you will see how the wavelength of light, the direction of illumination, and the spacing of lines in a diffraction grating can control the direction into which incident light is diffracted. What Gabor realized is that the diffraction grating that produces a given diffracted beam from a given incident beam is precisely the diffraction grating (i.e., fringe pattern) produced by two beams identical to the incident beam and the diffracted beam.