First, make sure you are capable of distinguishing and identifying ajnas by ear. Without those skills, the analyses on this site are too abstract to be sensible, and even if you think you understand them, you don't. You can polish up those skills by visiting the Beginner Track and by spending time on the Basic Ajnas section of the site, where you will find audio samples of all of the most important ajnas used in the maqam system. If you need to brush up on definitions, visit the Glossary, and if you need more clarity on how to read the content on this site, visit the How-to section.
You can follow the theory content of this site in one of two ways: 1. by following the entire analysis track in order: Start with the Introduction, which gives an overview of the theoretical approach presented here. After the Basic Ajnas section, you can proceed (by following the links at the top right of the page) to II. Jins Baggage, III. Unnamed Ajnas, IV. Common Pathways, and so on.
Or, 2. You can use the article "Maqam Analysis: A Primer" as your guide, which takes you through all the same basic content, but with more analytical writing and fewer audio examples. To my mind, the better course is to follow the article along with the full analysis track of the site, because the site contains many more examples that flesh out the analyses in the article.
The most important initial theory contribution of this site (and article) comes in the redefinition of the concept of jins, which in previous theory is simply a "tetrachord" (a 4-note scale fragment, following the model of the Ancient Greeks, whom Medieval Arab theorists copied), but in this theory is a center of melodic activity, defined not only by the intervals within a tetrachord or pentachord, but also by melodic patterns, mood, and tonicization. Furthermore, a close look at the repertory reveals that melodies "within a jins" (i.e. melodies sharing the same tonic emphasis, mood, and continuity) span beyond the traditional tetrachord boundaries of the jins. This idea is explored in the "Jins Baggage" section. A consequence of that new understanding is the discovery that there are ajnas in the repertory that have not been named in previous theory; they are identified here in the "Previously Unnamed Ajnas" section.
The second theory contribution offered is the idea that a maqam is best presented as a network of ajnas, rather than a scale divided into two parts. That view comes from the observation that songs and improvisations in a given maqam follow a set of common pathways among ajnas, an idea developed in the "Common Pathways" section of the site, and refined in the sections on "Alternate Pathways" and "Unique Pathways", which deal, respectively, with historical and geographical variation, and individual invention. It is all put together in the final analysis of Aruh Li Meen, and the consequence is the representation I developed on the Maqam Network Demo Site.