The electron contains finite negative charge. The same charges repel each other. What makes electron stable and why does it not burst? Is it a law of nature that the electron charge is the smallest possible charge that can exist independently? What is a charge after all? Is it like space and time or we can explain it in terms of some other physical quantities?
That's a great question!
Unfortunately, the only honest answer is "that's what we see in nature, with great precision and complete reproducibility." There is no deep theoretical understanding.
The more exotic form of your question is phrased in terms the self-energy of an electron, and it's a question that plagued Nobel Laureate Richard Feynman his entire life. He first tried to imagine that the field that emanates from an electron simply wasn't seen by the electron itself, in somewhat the same fashion that a woman standing on a high plateau cannot "see" her own height, since from where she stands everything around her is flat.
It didn't work. It did accidentally lead him to some methods for dealing with the problem that won him that Nobel Prize, but his own conclusions later in life was that he and everyone else had pretty much flat-out failed, and that the self-energy of an electron was still a mystery. It continues to be.
So why is it this problem so hard?
You stated it pretty well in your question: If like charges repel, then why doesn't an electron simply blow itself apart? After all, if you take a hundred small negatively charged object and try to push them together, the energy need just keeps increasing as you push them closer together. For an infinitely small point object like an electron, that energy goes to infinity! So, point particles and charges just do not play well together... yet half the charges in our world are composed of just such particles! And even the protons have their own version of the problem because of the three point-like quarks that are bounding around within them.
Here's a somewhat different way to visualize the self-energy problem. I like it because when I was a kid I liked to put my thumb over the open end of a water hose to see how far and fast I could the water squirt out.
The analogy works like this: It turns out that you can model an electric field remarkably accurately by simply picturing a positive charge as the end of a hose that is spewing out a fixed number of liters of water per second into a pool of water (space). A negative charge then becomes the end of another type of hose that sucks water in from the pool at the same rate. James Clerk Maxwell, one of the most flat-out brilliant physicists in all of human history, was one of the first people to notice this analogy. That is why "lines of force" are also sometimes called "flux" (meaning "flow") lines. Maxwell made good use this analogy to help him derive his famous Maxwell's Equations, by which he was able to unify magnetism and static electricity into a single unified theory of forces. It was Maxwell who first realized what light really is, by applying his own theory.
Back to the self-energy problem: Have you ever put your thumb over a pipe end that wants to spew out water at a fixed rate no matter how small the opening is? What happens is that the water speeds up and becomes far more forceful. A hose that gently drops water out when is opening that is several centimeters across becomes a tiny but amazingly intense fire hose when most of that opening is blocked. The slow flow becomes a micro-torrent whose speed is so high it will cut into soft objects.
Picture the size of that hose end as the size of an electron. If it's a big opening, no problem. You get the full flow without ever reaching extreme velocities.
But what if you start making the end of the hose smaller and smaller? It still has to produce the same number of liters per second, so just like when you put your thumb over the end of a big opening and try to block it up, the flow of water speeds up. The smaller the opening, the more extreme the speed up, e.g. cutting the outlet size in half causes the speed of the water to double, just so it can keep up. The same is true for the electron, just with the "strength" of the field replacing the "speed" of the water, and with "field lines" replacing the path of flow of the water.
So, if you shrink the size of the "exit hole" (whether the end of a hose or the size of an electron) down to a point, what happens?
Easy: The speed goes to infinity... which of course can't happen! Infinite speeds aren't even possible for water, which is limited by the speed of light. They would require unlimited energy just to approach the speed of light.
The situation is no better for point-like charged particles, which similarly must acquire field densities (think speed) that approach infinity. That in a nutshell is another way to understand a bit more visually why the self-energy is so tough.
So, with all that said, is is possible that someday there will be a theory that truly explains such things -- deeply explains them, in that special way the gives people reading it that cool little "wow, now I finally get it!" feeling as everything finally clicks together?
Well, not right now. String theory doesn't really bother with such problems, and in fact arguably makes such issues worse by picturing everything as tiny strings whose self-energy is pretty astronomical. Harari-Shupe Rishon theory is more of a nice organizational observation than a theory (think of the Periodic Table), but all it does is make all the point charges one-third that of an electron. I don't know if Garrett Lisi has ever tried to address the self-energy issue in that remarkable E8 theory of his, but it would at least provide a new angle on the problem that might shed some fresh light on it.
So, again: Great question, but alas, there's no one who can answer it yet! But who knows, someone reading this answer may be the person to solve that one someday -- why not? Yes, it's a very hard problem... but then most folks have no idea what they are truly capable of if they have a knack for physics, are truly interested, and are willing to work hard at it.
It's a very good question. The electron is described by a wave field which resembles a charge distribution, so it is natural to wonder why it doesn't repel itself and spread out all over.
However, the wave is not a classical wave but is quantized, i.e. the energy in a given vibration mode has to come in discrete bundles. One can count how many excitations are present and the number is an integer N > 0, the number of "particles" present. Also, the different vibration modes are identified with momentum.
A single-electron state is a "superposition" of N=1 excitations. The main reason to superpose is that any given mode extends over all space, but by superposing modes of different momenta one forms a "wavepacket" which is peaked in the center and then falls off to zero at greater distances, because the modes of different frequencies interfere destructively. The wavepacket concentrates the electron's charge into a confined area, and it seems like it should repel itself and grow rapidly wider.
However, this doesn't happen because of quantization and momentum conservation. The electromagnetic field works by carrying energy from one mode to another. One can think of it in terms of resonance: the electron field vibrations excite a (non-resonant) mode of the electromagnetic field, which does not travel far due to lack of resonance, but can still transfer energy permanently to another field which is resonant.
If there are multiple excitations (N >= 2) then the electron's vibration frequency is the sum of the individual mode frequencies, and there are many different ways to sum two frequencies to get the same thing. In other words, a two (or multi) particle state has many non-trivial resonances with itself and it can exchange momentum amongst its modes, allowing the packet to split apart and separate.
However, for an N=1 state, the only resonance of a given mode is with itself. The electromagnetic field (or any other mechanism) cannot transfer any energy between the different modes, hence they all stay the same and the single-particle state is stable.
This doesn't mean that the single-particle state is unaffected! It still excites the electromagnetic field and puts some energy into it, hence the energy of the vibrating electron field is larger than it naively appears because it is carting around the associated electromagnetic vibrations everywhere. The extra energy depends mostly on the electron's amplitude (not frequency) and hence looks like a change to the electron's mass, (because mass in quantum field theory is a measure of how the energy depends on amplitude). Therefore the electromagnetic field renormalizes the electron's mass, but isn't able to split the electron into pieces.
Of course quantization, which is the crucial piece here, remains a black box in this explanation....but hopefully it's better than nothing!
The electron is stable (from a QFT POV) because there is no particle with less energy (in the Standard Model) that it can decay into while also preserving all known conservation laws.
As for the "burst" part that question comes from a classical view of the electron which does not hold since we know from QM that electrons also behave like dual wave-particle entities. You could convert electron's mass in energy and calculate what would be the radius of a charged sphere (or spherical surface) with total charge $-|e|$ so that they would have the same energy but that would only give a classical radius. Near and inside that radius you would need QM or QFT to describe the electron. And from a QFT point of view we have not seen the electron behave like anything other than a point particle (when we observe it as a particle) no matter how close we "probe" it.
Regarding what charge is I d describe it as an observable, measurable and scalar quantity. (It is also conserved because of the U(1) symmetry of E/M.) Maybe not the best answer but that is how I approach that issue.
If we calculate the energy required to assemble an electron from a thinly spread out cloud of charge as e, then e is almost equal to the rest mass of the electron. (This is taking into account the experimental size of the electron from collision experiments and uniform charge density in the electron). Though this information does not explain the stability of the electron...gives some picture i.e. It is concentrated energy. The spin and the magnetic fields can be the primary binding reason - leading to the quantum wave theory.