Louis de Broglie said that electrons are waves. But how they repel each other?
Electrons aren't waves or particles. Instead they are excitations in a quantum field called the electron field. The interactions between electrons are described by the equations of motion of the electron field.
However de Broglie was correct in the sense that if we consider an isolated electron this can be approximately described as a state of the electron field called a Fock state, and this is basically a plane wave. So for an isolated free electron it is a very good description to consider it as a wave. There is more on this in my answer to What is a subatomic particle? if you're interested in pursuing it.
However if we have two electrons then the state of the field is not just two Fock states superimposed. If it were then the electrons wouldn't interact so there would be no repulsion between them. In fact we can't solve the equations for the field state directly so we use a technique called perturbation theory to approximate what happens. If we consider two electrons heading towards each other, interacting and then heading away from each other. The calculation is done by summing up a series of calculations represented by Feynmann diagrams. You've probably seen Feynman diagrams like this:
This shows the two electrons interacting by exchanging a virtual photon, but be careful about interpreting this literally as it's just a representation of an integral called a propagator. No virtual photon actually exists. What we are doing with these diagrams is calculating how the two Fock states interact and in the classical limit this gives us the force between the two electrons.
Electrons are point particles in the standard model of particle physics, as seen in the table. They are quantum mechanical entities: when detected they leave a particle footprint within the Heisenberg uncertainty principle.
Here is an electron in the hydrogen bubble chamber, it leaves a consecutive macroscopic footprint of a charged particle moving in a magnetic field. The dots are small scatters off the hydrogen atoms which compose the bubble chamber.
Where is the wave nature? At the original interaction vertex when the K- hit the hydrogen atom and transfered enough energy to give kev energy to the electron. The probability of this happening is the square of the quantum mechanical wavefunction for the interaction: K- + hydrogen atom ----> e- K- H+ .
The single electron double slit experiment shows both the individual footprint, and the wave nature of the interaction : "electron scattering of double slits" . The single electrons look random on the screen. The accumulation which is a probability distribution, gives the interference pattern of a wave nature.