De Broglie explained this beautifully in his seminal Nobel Lecture of 12th December 1929:

[…] I was attracted to theoretical physics by the mystery enshrouding the structure of matter and the structure of radiations, a mystery which deepened as the strange quantum concept introduced by Planck in 1900 in his research on black-body radiation continued to encroach on the whole domain of physics.

[…] For a long time physicists had been wondering whether light was composed of small, rapidly moving corpuscles.  This idea was put forward by the philosophers of Antiquity and upheld by Newton in the 18th Century.  After Thomas Young’s discovery of interference phenomena and following the admirable work of Augustin Fresnel, the hypothesis of a granular structure of light was entirely abandoned and the wave theory unanimously adopted.  Thus the physicists of last century spurned absolutely the idea of an atomic structure of light.  Although rejected by optics, the atomic theories began making great headway not only in chemistry, where they provided a simple interpretation of the laws of definite proportions, but also in the physics of matter where they made possible an interpretation of a large number of properties of solids, liquids, and gases.  In particular they were instrumental in the elaboration of that admirable kinetic theory of gases which, generalised under the name of statistical mechanics, enables a clear meaning to be given to the abstract concepts of thermodynamics.

[…] Some thirty years ago, physics was hence divided into two: firstly the physics of matter based on the concept of corpuscles and atoms which were supposed to obey Newton’s classical laws of mechanics, and secondly radiation physics based on the concept of wave propagation in a hypothetical continuous medium, i.e. the light ether or electromagnetic ether.  But these two physics could not remain alien one to the other; they had to be fused together by devising a theory to explain the energy exchanges between matter and radiation – and that is where the difficulties arose.  While seeking to link these two physics together, imprecise and even inadmissible conclusions were in fact arrived at in respect of the energy equilibrium between matter and radiation in a thermally insulated medium: matter, it came to be said, must yield all its energy to the radiation and so tend of its own accord to absolute zero temperature!

[…] “A wave must be associated with each corpuscle and only the study of the wave’s propagation will yield information to us on the successive positions of the corpuscle in space”.

In conventional large-scale mechanical phenomena, the anticipated positions lie along a curve which is the trajectory in the conventional meaning of the word.  But what happens if the wave does not propagate according to the laws of optical geometry, if, say, there are interferences and diffraction?  Then it is no longer possible to assign to the corpuscle a motion complying with classical dynamics, that much is certain.  Is it even still possible to assume that at each moment the corpuscle occupies a well-defined position in the wave and that the wave in its propagation carries the corpuscle along in the same way as a wave would carry along a cork?

[…] Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time.  The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted.  It has thus been possible to predict a whole group of phenomena without their actually having been discovered.  And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.