Light was known to be a wave, and yet it was a particle as well. The momentum of one photon is p = \dfrac{h}{\lambda}, where wavelength, a quantity of a wave, and momentum, a quantity of a particle, are related by a single formula containing h.
De Broglie read this relation the other way round. Rewriting it as \lambda = \dfrac{h}{p}, he gave a wavelength to things that had been taken for particles.
Put another way, it is not only light that has the properties of both a wave and a particle. An electron, an atom, a baseball: whatever has momentum has a wavelength. This wave is called a matter wave, or a de Broglie wave.
For a particle of mass m moving at speed v, the momentum is p = mv. The wavelength of the matter wave comes to \lambda = \dfrac{h}{mv}.
The faster it moves, the shorter the wavelength. The relation being inverse, doubling the speed halves the wavelength. Something at rest has an infinitely long wavelength, which is to say it sits nowhere in particular.
Since h is a number as small as 6.6 \times 10^{-34}, everyday speeds and masses give a wavelength that is outrageously small. Whether something can be seen as a wave comes down to how that smallness is made up for.
Accelerate an electron through a voltage V and the potential energy it loses becomes kinetic energy just as it is. That is eV = \dfrac{1}{2}mv^2.
Turning this into p = mv gives p = \sqrt{2meV}, so the wavelength comes to \lambda = \dfrac{h}{\sqrt{2meV}}. Being inverse in the square root of the voltage, raising the voltage fourfold halves the wavelength.
An electron accelerated through 100 V has a wavelength of about 1.2 \times 10^{-10} m. That is just about the spacing at which atoms sit, and this agreement leads into what follows.
The wave nature can be seen only when the wavelength is about the size of the structure in its path. Sound bends around a wall because its wavelength is about the size of the wall.
Throw a 150 g ball at 40 m per second and its wavelength is about 10^{-34} m. That is 24 orders smaller than the size of an atom, and there is no gap anywhere of that size.
The 10^{-10} m of an electron is the size of the spacing between atoms in a crystal. That is why a crystal stands in for a diffraction grating, and the wave nature appears in a form that can be seen.
Send an electron beam into a crystal and it comes back strongly only at particular angles. The atoms sit in a regular arrangement, so the angles at which waves reflected from neighboring planes reinforce each other are fixed.
Writing the spacing between the planes as d and the angle of reflection as \theta, the difference between the two paths is 2d\sin\theta. They reinforce when this is a whole number of wavelengths, that is, when 2d\sin\theta = n\lambda.
Davisson and Germer measured this and confirmed that the \lambda that came out agrees with \dfrac{h}{\sqrt{2meV}}. That an electron is also a wave became a measurement here.