Nowadays we know it is a consequence of the wave property of light. But ironically it was discovered by Newton who held the particle point-of-view of light. So how did he explain his discovery?
"Newton rings" in thin plates were discovered not by Newton but by Hooke, and not in 1717 but in 1664 (Boyle described a similar phenomenon in soap bubbles at about the same time). Hooke reported his experiments in Micrographia (1665), from which Newton learned about them, "in which book he hath also largely discoursed of this… and delivered many other excellent things concerning the colours of thin plates", as he acknowledged in his Discourse of Observations (1676). But when the Discourse was converted into Opticks and published after Hooke's death the mention was taken out, hence the successors learned of them as "Newton rings", see Encyclopedia's summary. In fairness, Hooke only reported the picture qualitatively, whereas Newton performed detailed measurements so precise that Young could later derive wavelengths of various hues from Newton's data to high accuracy.
Despite the commonly repeated assertion that Newton's optics was corpuscular (and this was likely his initial intuition) his considered position was actually a hybrid of corpuscular and wave theories (reminiscent of the later de Broglie's pilot wave theory), although he presented the wave part as "hypothetical". The hybridization already appears in his optical manuscripts of 1670-s, when he had to accomodate the partial reflection and refraction in thin plates and diffraction (discovered by Grimaldi in 1665), see Brewster's survey at the Newton Project. As a result, Newton's corpuscles were not elementary but had what we now call internal degrees of freedom, vibration modes, which he called "fits" and which interacted with the all-pervading "luminous ether". Here is a detailed explanation from Preston's Theory of Light (freely available online):
"The existence of both reflection and refraction at the surface of a transparent substance presents at first sight a great difficulty in the emission theory, for it is not eaay to conceh-e how the same surface may at one time reflect and at another refract an impinging molecule. To meet the difficulty Newton was led from his observations on the coloured rings of thin plates (chap. VIII) to endow the luminous corpuscles with periodic phases or fits, as he terms it, of easy reflection and easy transmission, so that sometimes they are in a condition to be reflected, and sometimes in a condition to be refracted at a transparent surface.
"To communicate these fits to the luminous corpuscles he imagined all space to be filled with an all-pervading medium or ether. The luminous corpuscles, on striking a reflecting or refracting surface, excite waves in this ether which overtake them at regular intervals, and assist or oppose their motion periodically, so that at any new surface they are refracted or reflected according as the wave assists or opposes the corpuscle. The element of periodicity thus so ingeniously introduced, and which is so fundamentally involved in a wave motion, we should naturally expect to be independent of the angle of incidence. However, to reconcile the theory with his observations on thin plates, Newton found it necessary to suppose the length of a fit to vary as the secant of the angle of incidence, and it does not appear easy to account for such a law."
And here is Newton himself in book II of Opticks applying his hybrid theory to explain the rings in thin plates:
"Every ray of light in its passage through any refracting surface is put into a certain transient constitution or state, which in the progress of the ray returns at equal intervals and disposes the ray at every return to be easily refracted through the next refracting surface, and between the returns to be easily reflected by it... This is manifest by the 5th, 9th, 12th, and 15th observations (coloured ring). For by those observations it appears that one and the same sort of ray at equal angles of incidence on any thin transparent plate is alternately reflected and transmitted for many successions accordingly as the thickness of the plate increases in arithmetical progression of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, etc."
Strictly speaking, Newton only needs coordinated "fits" to explain the rings, and mathematically his explanation is essentially equivalent to that in wave optics. But the nature of this coordination remains mysterious without the ether. To cover his bases Newton in Opticks first professes the familiar hypotheses non fingo pretense:"What kind of action or disposition this is or whether it consists in a circulating or a vibrating motion of the ray, or of the medium, or something else, I do not here inquire". But for those "averse from assenting to any new discoveries but such as they can explain by a hypothesis" he provides the rest of his account phrased into rhetorical questions (the idea dates back to Hypothesis Explaining the Properties of Light, 1675):
"And the vibrations or tremors incited in the air by percussion continue a little time to move from the place of percussion in concentric spheres to great distances. And in like manner, when a ray of light falls upon the surface of any pellucid body, and is there refracted or reflected, may not waves of vibrations, or tremors, be thereby excited in the refracting or reflecting medium at the point of incidence and continue to arise there, and to be propagated from thence... and are not these vibrations propagated from the point of incidence to great distances? And do they not overtake the rays of light, and by overtaking them successively, do they not put them into the fits of easy reflexion and easy transmission described above?"