Can you explain why it is considered a point-particle (unlike the proton), that is: having no spatial extention? Doesn't mass have to occupy space anymore?
The electron is an elementary particle, and thus a quantum mechancial entity, of the standard model of particle physics.
The proton is composed out of elementary particles shown in the link in a table.
Quantum mechanical entities are described by solutions of quantum mechanical equations, Dirac for the case of the electron, plus the field theoretical solutions of the lagrangian of the standard model linked above. These solutions give the probability of finding the electron at (x,y,z,t). Depending on the boundary conditions of the problem at hand, this probability might be constrained on a point, if the dimensions are macroscopic, or display wave like characteristics.
In a recent answer user AnnaV showed that it has even been filmed, why do they think it must be definitively smaller than 1/10^-16 cm? How can such a lot of mass be squeezed into a vanishing point?
It is the way the mathematics of the standard model works out. The particles in the table are entered as point particles in the lagrangian, the mathematics is cranked, and predictions for the behavior of the electrons are made. These predictions work to a great accuracy, as demonstrated by the four LEP experiments..
That is the why, because the predictions of having the electron a point particle fit the data.
Now theory advances and predicts that they are not point particles but have a one dimensional extent, at values smaller than 10^_33cm they are strings. With the new energy of the LHC it may be possible to start seeing the string effects. It may also be necessary to wait for higher energy machines to see whether elementary particles are really strings in many extra dimensions. At the moment the definition rests on the standard model of particle physics and its successes up to now.
Best Answer
Scattering experiments can be used to determine the size of a particle. The results for an extended object are different than that of a point particle. But all of these scattering experiments depend on getting the probe particle "close" to the scattering object. In the case of electrons, that means launching the probe with enough energy to overcome the Coulomb repulsion ... and get "close". How close depends on the energy of the probe particle. But there's a limit on the energy that can be given to the probe, so there's a limit to how "close" the probe can get. Consequently, we can't know what happens at distance shorter than some value.
The best scattering experiments done on electrons to date show a scattering pattern identical to that of a point particle.
That doesn't mean that it actually is a point particle, it just means that to the best of our knowledge it is. No experiment to date, scattering or otherwise, has shown any sign of a size for the electron. Our theory and our experiments are both happy ... for now ... with the notion that the electron is (more accurately: behaves exactly like) a point particle.