Though Bohr was able to provide some remarkable results, they did not seem to be able to be extended beyond where they reached in the first short years. People worked with this for some time but became uncertain as to where to go next.
Then in 1923, as part of his Ph. D. Thesis, Louis de Broglie (he was of the French aristocracy - hence the title "Prince") argued that since light could be seen to behave under some conditions as particles (photoelectric effect) and other times as waves (diffraction), we can also consider that matter has the same ambiguity of possessing both particle and wave properties. He suggested that the quantization observed everywhere experimentally and asserted in an ad hoc fashion in Bohr's theory arose from this wave property of matter. We know that a guitar string can flop around in any old way when left sitting on the table. But when we tie down its two ends and apply a specific tension, it vibrates with only certain specific frequencies - a fundamental and overtones of that fundamental. By confining the string, its wave properties become quantized and only a select few frequencies are ever observed for the vibration of that string. We can change sounds by changing the confinement - pressing on a fret in order to make the string effectively shorter - or by changing the tension. But once its confinement conditions are specified, it has a very specific set of vibrations from which it can choose once stimulated by its being plucked. de Broglie suggested that when an electron is confined in an atom, its wave properties are quantized also.
He arrived at an expression for the wavelength associated with a given particle by reasoning in analogy with photons. The energy of a photon is related to its frequency through Planck's constant. But Einstein's special theory of relativity also related its energy to the speed of light. By equating these two relations and converting its frequency to a wavelength an expression for wavelength in terms of Planck's constant, mass, and velocity was determined. de Broglie's leap of faith was in asserting that this same expression should apply to particles. It is instructive to get a feel for the wavelength of various particles you might encounter.
These ideas fell into the hands of others who tried to see if it could be the basis for a new theory of matter.