How can Neo, Morpheus, Trinity etc. change/break the Matrix's rules (jump so far, fly, or move fast)? Do they have ability to "compile" their thoughts into the matrix code on the fly? How can they have any influence on the Matrix at all?
Two other questions are relevant here: Did the machines use humans' brains as CPUs? and Was executive meddling the cause of "humans as batteries" in The Matrix?
Originally the Matrix was going to be software running on human-brain hardware. This scenario obviously makes control of the parameters of the matrix much more plausible. By achieving the correct mental state, the redpills could alter the simulation running in their heads consciously, in the same way as lucid dreamers report having better control of their dreams.
While the script eventually used the "batteries" idea, none of the rest of the plot and capabilities were changed along with this - the matrix is still presented as a form of artificially-induced group hallucination. It seems the Warshowski's intent was that there is still a strong connection between the brains of the operatives in the matrix and the world they perceive.
Consider that the matrix is not just an uncontrollable simulation but responsive to input - relying on humans' natural ability to dream and imagine the world around them. The simulation can't possibly be an atomic-level physics model of the entire city; that would take far more work than actually building it! Instead it must be, like all simulations, "good enough", relying on the human brain's strong pattern-recognition abilities to fill in the gaps. The input has to be two-way for the matrix to work - the matrix is affected by the human mind, as well as the other way around. (Otherwise each person would be in their own personal matrix, not interacting with the same world the other people see.)
Presumably the system includes 'error-checking' systems like the client code in online computer games, which corrects the difference if the individual person is moving in a way not allowed by system as a whole.
But the redpills have been disconnected from any such error-checking mechanism when they left the matrix. They don't reconnect to it, since they're re-entering the matrix by hacking anyway. And they're consciously aware of this immunity. So the redpills can "fill in the gaps" by assuming the rules work the way they want - again, in the same way a lucid dreamer can control the dream.
Do you believe that my being stronger or faster has anything to do with my muscles in this place? Do you think that's air you're breathing now?
The matrix still has to feed the inputs into your brain and that is pushed back into the matrix. So a lot of what they do is based on how fast or slow they input into their brain. Other parts are suppession of biofeedback and tricking the matrix into changing their mass.
The basic concept as I understand it is that the Matrix is the last word in "social networking". The machines provide a basic framework, and a measure of control over the entire process, but the machines need the humans to "buy in" and live in this world as if it were reality. The humans are the primary source of input, not the machines; they interact with each other much more often than with artificial entities like Agents. The machines simply provide an initial state with enough detail to make a human mind suspend disbelief. It works for the sole reason that the humans choose to believe in this world; the collective consciousness becomes self-reinforcing through the billions of inputs that validate the sense data the Matrix is feeding them as reality, and thus subject to reality's rules. Once they do, the machines simply need to police it and patch it as necessary to maintain the illusion.
Because of this basic property defining the Matrix's reality (the Architect calls it a "fundamental flaw" that is tolerated because it functions where his original perfection did not), humans who know the Matrix isn't real are feeding in their own input incorporating that knowledge. A human who thinks the Matrix is real will "know" that they weigh a certain amount, and that their strength is based on their muscles. A person who knows the truth will know that in the Matrix, if they don't want to weigh anything, they won't, or that if they want to jump from one building to another, they can, regardless of the distance.
It's still difficult for most of the unplugged to truly disengage from the Matrix's rules because at some level, some part of them still chooses to believe that there is a fundamental level of reality to it. As such, they can bend or break certain rules, but some things, like death, are just so permanently ingrained that they're inescapable. Neo, somehow, transcended even that, and so virtually all of the programmed rules only apply to him when he wants them to, to appear normal. He can even violate fundamental properties of physics as programmed by the machines; the simulated matter of his body can occupy the same space as other matter, like Agent Smith's body, or Trinity's.
The idea behind the breaking of limitations of the Matrix is that the limitations do not exist at all, we impose these limitations upon ourselves based on our idea of what is "true". Legends tell of Eastern monks forgoing food and water, accepting that we are simply addicted to these things, and they are not required as much as we believe. In the same way we accept physics as true, because we can not comprehend it not being true.
This is the importance of the jump. While the mind is still tethered to notions of human capability, they are incapable of making the jump. However the fall and subsequent landing helps break the notion, seeing as you don't die. As one breaks their concepts of what can be true the rules break down.
The red pill has nothing to do with it, people who are still jacked in can break through these rules, such as the children in the Oracles room, and the runner from Animatrix. It's important to note that most of the people at the Oracle were children, and they note that most of the time they free a mind when they are much younger then Neo. This is likely because as children we are less accepting of the "truth" and easier to accept that what we understand is false.
As for Neo, he has super user access. Whereas the Matrix does impose rules for basic governance that allows for all levels of human capability (jumping really far, moving fast, etc) Neo can temporarily suspend or is exempt from such rules. This is mostly shown in his ability to fly, based on the fact that no human can fly regardless of how strong or fast they are.
The same way any malicious program exploits the systems it infects. By finding the input which provokes the desired output or system state. In the real world this can be achieved in a myriad of ways; you can flood a system with requests until it crashes, you can inject malicious code through unfiltered input, you can hijack data from a legitimate sender, alter it to suit your purposes and send it on to it's intended recipient, etc. The weaknesses of the system depends on its implementation and the freedom it gives its users to interact with it.
If you've got a tightly restricted program that only accepts a very small number of possible inputs, and you've accounted for every possible form that input may take, you're generally pretty safe. But the more complex the system gets, the harder it becomes to guarantee its behavior.
Let's say you have a program that takes as input two natural numbers between 1 and 2. It then adds these two numbers up and returns the result. This really simple program has 4 possible inputs: (1,1), (1,2), (2,1), (2,2). It's not hard to test how each of these four inputs will make the program behave, you can even sit down and test it for yourself manually in a matter of seconds. So you can say that's a pretty secure program.
But let's say the program needs to be a bit more complex. You want to allow every natural number as input. Suddenly you have an infinity of possible inputs (restricted by the input interface, of course) to test. You say that's probably okay, we can test a few thousand numbers and see it checks out as it should. Hell, let's do a few million just to be sure. So you release your program into the wild and someone inputs 4,294,967,295 + 1 and suddenly your program crashes with an integer overflow and returns an exception containing the stack trace of your whole framework. You couldn't foresee that, the system was too complex to test every possible input.
The Matrix is a virtual reality simulation so complex that it allows several billion concurrent users to interact with it in such a way as to be unable to distinguish between it and the real world. Our bodies are absolutely littered with sensors and actuators, every square centimeter of skin interacts with our surroundings in some way. If we assume the Matrix reconstructs that experience accurately, then the input our bodies, and therefor our minds, injects into the system every second would be enormous. If they cut corners and simplify to simulate approximate experiences then that only makes the potential attack area even more error prone and open for exploitation.
As Neo becomes more aware of the Matrix and how it works, he begins to treat it less as a set of rules that define him, and more as a system that can be hacked.
"You think that's air you're breathing?" asks Morpheus, implying that breathing is just another input / output process (no pun intended).
"There is no spoon. [...] you'll see that it's not the spoon that bends, it's only yourself" = you are not interacting with a real object, you are changing the input your mind gives to the program in order to exploit a weakness.
The Matrix was hacked.
Even in real world, malwares use loopholes to go against rules of Operating System. The only difference is that they aren't intelligent objects to find, create and use loopholes at the runtime.
Neo, Morpheus, Trinity, Smith etc as intelligent objects of the system actually override rules subroutine of The Matrix. Neo & Smith went too far against it because they were able to understand the system with far greater details (Neo was awarded those capabilities by Architect & Smith was a real virus evolved by chance).