According to this question and Monty's answer:
Neo has within him a portion of the Source Code of the Matrix.
Let's suppose that it's true. But how is he, a human being, able to have the source code within him?
When you talk about a person who is within the Matrix, there are a few things that you can be talking about:
In this case, we're talking about Neo's mind and/or avatar having within them the Matrix code. This trope (the idea that someone's mind can hide code or lengthy keys) is present in other fiction as well. Since most humans' minds are fully within the Matrix and interact with its constructs regularly, it's not too much crazier to try to understand that "the One" is able to interact with those constructs on a further level back, due to having a higher privilege level and more information (the source code) within the Matrix. In order to access this higher privilege level, however, he has to go through the trials that are shown in the first movie.
There is another example of humans being affected by the Matrix source code. In The Matrix Reloaded, former Agent Smith replicates his code into the minds of humans and is able to escape out of the confines of the Matrix.
Who: The Architect would place the source in a single individual's Matrix persona in every cycle. This person would then go on to be "The one".
What for: He was forced into doing this because the perfect Matrix eventually grew unstable and ended up in a crash losing all human minds hooked up to it. As a solution to this, rather than the anomaly being unknown and random, he made the Anomaly controlled by introducing it himself. It's similar to the difference between letting an old building collapse and endanger everyone, and evacuating everyone and demolishing the building on purpose. I hope you get the analogy I'm trying to bring in here.
I dont remember any passage from the movie, but the Architect said:
"Your life is the sum of a remainder of an unbalanced equation inherent to the programming of the matrix. You are the eventuality of an anomaly, which despite my sincerest efforts I have been unable to eliminate from what is otherwise a harmony of mathematical precision. While it remains a burden to sedulously avoid it, it is not unexpected, and thus not beyond a measure of control. Which has led you, inexorably, here"
So he has no place normal in matrix, more complex relationship than other.
It's maybe because his mind is connected to the source but it's maybe because Neo is not human. The matrix is placed to hide the reality. Like we know, the things are not what they are supposed to be. The story may have several levels of reading, including one hidden.
Let me develop. The desire for freedom and rebellion in humans is too strong to be contained. It was necessary that the Machine find a ploy to keep humans under domination. If the rebellion can not be avoided, the machines must then create the illusion of a rebellion, which takes place in the matrix. A fake prophecy and a fake messiah, and a fake reality to keep humans prisoners of machines.
To answer your question exactly, "how is a human being able to have the code within him?", remember that NEO is born artificially has a human battery for the big computer and that he was linked to it since he was born.
That is why you see many NEO in the Architect room, every time he is reborn he even look the same as the previous time. The machine and the humans are both part of the same entity. One can't exist without the other anymore.