A proton is stable because of the strong force between quarks, which is not there in electron. So what's the reason for electron's stability?
As far as we know, electrons are fundamental particles and have no internal structure or components. Also, an electron cannot decay into other particles (unless it has a very high kinetic energy) because there is no lighter charged lepton for it to decay into. It can, however, annihilate with a positron to produce gamma rays.
An electron is an elementary particle in the standard model of particle physics. . The table axiomatically assumes that elementary particles are point particles in the QFT of the model, i.e. have no constituent parts.
Depending on the quantum number conservation rules and if there exist consistent lower mass particles to decay to, elementary particles can decay, even though they have no constituents.
The electron has the electron quantum number , and the only lower mass particle is the electron neutrino, and the photon with zero mass is available, ( at least two for momentum conservation in the center of mass) but both are neutral so charge would not be conserved. Thus the electron is point like and stable, as far as our data and the theory that fits these data are concerned.
A proton is stable because of the strong force between quarks, which is not there in electron
So you suggest a proton must rip itself apart, or has the ability to, as it is made up of quarks. But do the quarks also need to rip themselves apart? Its the same for the electron. We consider quarks and electron, both to be elementary - experimentally and theoretically. There is nothing in them to make them rip themselves apart.
Besides there are other deeper reasons, which other authors have remarked upon.
"Why" is more of a philosophical question instead of physics one.
From our observations and experiments in the particle world it looks like two types of particles don't decay:
massless particles (photon, gluon) - they simply don't feel "time".
particles that can't decay without breaking some known conservation law (like electric charge or mass).
All others are known to decay into lighter particles until one gets into one of the two cases above.
The proton cannot decay into anything while still conserving electrical charge, baryon number and mass. All other known particles are either heavier, wrong baryon number, or electrical charge.
Electrons are limited in the same fashion - electrical charge, mass and lepton number are all conserved (as far as we know) properties.
Then again, we are not absolutely sure that electrons and protons don't decay. A lot of effort is made searching for decay modes for both the proton and the electron and their half-life is (as of now) limited to not less than some mind-boggling number of years like 10^35.
Observing a proton decay will invalidate some of the conservation laws as we know them.
A proton being a bound state of quarks doesn't change the picture. We don't know if the quarks are stable if unbound, they may as well not be, or at least the down quark may be able to decay into up one. We cannot separate them for long enough to see.
But, the bound state being stable while free particles unstable is pretty much known in the atomic nuclei. Neutrons are prone to beta-decay when free and pretty much stable when bound in a stable nucleus. The bound state has lower enough mass than its constituents to make the decay impossible.
The electron is a point particle as far as physicists know. If you apply the electrostatic self energy formula for a charge distribution to a point particle, you will find infinity. The only conclusion we can draw from this is that we cannot consider an electron as a static charge distribution.
An electron is not point-like, as a matter of fact. Its "size" is state-dependent. Anyway, if you puch a still free electron, you will get an excited final state - a moving electron plus soft radiation. Looks like it was not "free", but coupled permanently within the EMF oscillators, to say the least. This coupling smears the elastic (non destructive) picture (photo) of a "free" electron. Point-like picture is inclusive one - it includes all possible excitations during observation.
According to some theories (as Cosmas Zachos rightly commented), the electron can be torn apart. If you consider the electron made up of three more elementary particles (each with a unit charge of -1/3, so they actually are antiparticles) then it's also easy to see how they can change identity in high energy interactions. For example, an electron can interchange its constituents with a quark or neutrino to transform into a quark (while a quark can turn into an electron).
This shifts the question to the more elementary particles though. Why should a charged point particle be stable? First of all, it should be noted that all (electric) charge is ultimately made up from elementary charges. So it doesn't make sense to ask why they can't be subdivided further. They are just elementary charges, not made up of smaller charges. The infinite self-energy question isn't a question. This energy simply is not there.
Secondly, it could well be that the charges are not point-like. In a quantum theory of spacetime, it might well be that the pointy structures are some strange, very small (maybe Planck-sized) distortion of higher-dimensional space. It could be that this curling up of space (within the large, global structure of spacetime as described by general relativity) is all that a particle is. To say that a charge is present in this small curled-up space would be unnecessary, superfluous. Like in string theory charge is represented by the vibration of strings, which themselves contain no charge.
The simplest answer is: because it's held together by its own gravity. In fact, we could take this one step further: it actually is a gravitational soliton; nothing more, nothing less, and that there is nothing there to fly apart.
Here is a simple exercise that will help focus attention on the issue: write down an axially-symmetric solution to Einstein's field equations that has the same mass, same electric charge and same angular momentum as an elementary fermion - treating its spin as internal angular momentum: specifically the electron. What is that solution?
Answer: a naked Kerr-Newman singularity.
Now, having determined that to be the case, this raises the following question: since electrons (and more generally, fermions) are described by the Dirac equation, then shouldn't there then be some kind of close, and deep, correspondence between solutions to the Dirac equation and Kerr-Newman solutions? If what I said is true, then there must be, and this correspondence will then provide witness to the assertion made.
Well... actually, there is: The Dirac - Kerr-Newman Electron.
It's one of those "well-known" results that's been hanging around in the attic collecting dust, and hardly anyone's gone up to take a close look at it - partly because of the implication - and nobody's really figured out what to do with it. I'll cite the key points in the abstract:
We [...] show that the Dirac equation may naturally be incorporated into Kerr-Schild formalism as a master equation controlling the Kerr-Newman geometry. As a result, the Dirac electron acquires an extended space- time structure of the Kerr-Newman geometry - singular ring of the Compton size [...]
The rest you can read there in the link above. Do a web search on ArXiv for "Kerr-Newman Electron" and you'll find more recent material published on the matter. It's an area of active research.