Understanding mathematical texts
This isn't a complete answer by any means.
A couple of weeks ago there was a conference based on the work of William Thurston. There were several references made to an idea he used (I believe) of having levels of understanding. When you first meet something, you can read the theorems, and get a first level of understanding. But as you come back to it, in different contexts, seeing it from different points of view etc, you gain more and more insight.
In my experience learning mathematics is a lot like learning a language. You need that basic vocabulary, but in order to really have a conversation you need a deep understanding of what all the words really mean and how they fit together and interact, and all the subtleties therein.
Once we are proficient at a language we no longer worry about what each individual word means, rather we understand sentences as a whole. So now when someone says "How's it going" we take that at one piece of language, rather than three separate words.
For me mathematics has been the same. While in undergraduate studies I would always need to look at all the pieces of something to know what's going on. But in my MSc year I started noticing that I was starting to think about the areas that I studied in a much broader sense. Rather than having to drudge through all the details in my mind I could call up the relevant parts and leave the other bits alone, confident that they were there somewhere, but ok with not having them at the front of my mind because I knew how it all fitted together. This I guess is the type of understanding that you are asking about here.
So here is my answer to your question: This understanding comes from good old-fashioned hard work, repetition and lots and lots and lots of working through examples and approaching the ideas from different viewpoints and directions. It is only in my fifth year of university mathematics that I have even started to feel I have started to have this type of understanding, mathematicians spend their whole lives sharpening it and getting better. But there is no great secret, just like learning a language you start with the basics and then practise, practise, practise.... and one day you may find yourself writing poetry in this new language.
I think what Poincare calls "certain order" can also be called the mathematical idea (behind the subject you are studying). To understand a mathematical idea, the following items are important:
A mathematical idea is a dynamic creature and it continually evolves according to its applications. Take for example "continuity". It started as a notion for functions from $\mathbb{R}$ into $\mathbb{R}$ and its definition has extended to every function between two topological spaces. In fact, the main motivation for development of topology is to address the idea of "continuity" in its more general sense.
One should look for the preceding challenge that has raised the necessity for certain theoretical explanation and subsequently gave birth to the new idea. For example, in Euclidean geometry all curves (and lines) are implicitly assumed to be continuous curves. For instance, when we consider two arcs meeting each other at an intersection point we implicitly assume continuity. To see how continuity is used, note that the equation $x^2 = 2$ has no solution in rational numbers (irrational number did not exists at the time), but it is assumed that the curve $y=x^2 -2$ intersects the $x$-axis in two points. This conceptual challenge and its solution (or theoretical explanation) gave rise to several fundamental ideas of mathematical analysis such as continuity, real numbers, etc.
For understanding a mathematical idea, sometimes it is helpful to clarify the relationship between the idea at hand and other mathematical ideas. For example, to understand why groups are defined the way they are defined, considering groups as the symmetries of a geometric object is extremely helpful. surprisingly, this realization of groups as symmetries of geometric objects not only helps beginners to understand groups easily, but it is also a fundamental idea used by experts in group theory. In fact, the subject is called geometric group theory.
Never expect you understand a mathematical idea fully at the first encounter. It takes time to get a feeling about an idea. Therefore you should consider the learning of the formal definition and basic properties of a mathematical notion as the first step and you should develop and further your understanding gradually. I remember, first time I read the definition of a category, I memorized it but I had no feeling about it. However after a while, the definition of a category was so natural for me that I could not imagine another definition for it.
A useful practice to understand a mathematical idea (or subject) is to discuss it with others or even better give an informal (or semi-formal) presentation about it. Some people even teach a course to extend and deepen their understanding of a subject.
These are fundamental questions that how we understand something and how we can facilitate our understanding of a subject (in mathematics or other fields). So the above items are just some basic comments.