###### 1) Know that Hashem's first title...
This section is based on the idea that the lower sefirot are vessels and access points to the higher sefirot.
Consider the example of "love". Love can be understood to be an abstraction: We can talk about "love" without referring to any particular example of love. For our purposes in this example, let's think of "love" in this abstract sense as in the middle layer: Y-H-V-H or Tiferet. This is the idea of love, but not any particular love.
Now, let's imagine that instead of talking about love in general, you think of someone you love. When you hold them in your mind and think about your love for them, you can feel a concrete love, a differentiated love: the love of a particular person. You could get even more concrete: There is your love for them in general, and there is the love you feel right now (if you feel it now). This right-now-love is even more concrete and specific than your love for that person over your whole life. And that love – your love for a particular person – is more specific/differentiated/concrete than the idea of "love" in general, without any particular example.
For the Kabbalistic symbolism, your concrete love is אדנ"י, while the abstract idea of love is יהו"ה. Each instance of your concrete love is a gate, an access point, to love in the abstract. In fact, the only way you can access the meaning of love is through the particular: You can't really understand what love is without an example of love. That's why R. Gikatilla says that you can only access יהו"ה through אדנ"י.
###### 2) Everything below this is the world of division...
In Kabbalistic symbolism, we imagine that the higher and more abstract levels create or emanate the lower more differentiated levels: If there was no all inclusive abstract idea of love (Tiferet), then there could not be any concrete examples of love (Malkhut). In this way, Tiferet creates or emanates Malkhut: Each concrete example of love "emanates" from the general idea of love.
This is in some ways the opposite of how we think today: We think that the idea of love is an abstraction that arises from concrete examples of love – We have concrete loves (Malkhut) and so we come up with the general idea of love (Tiferet). But medievals often thought about it the opposite way: Since love exists as a universal abstract entity (the idea of love in Tiferet), so it is possible for you to manifest a particular instantiation of it.
Once you've made sense of this way of thinking, pay close attention to how R. Gikatilla is reading the verse. Reading the gan eden verse this way is a hallmark of Kabbalistic symbolism. Understood this way, the verse appears hundreds (if not thousands) of times in Kabbalistic literature.
###### 3) Know that this title Adonai is a treasury...
This section applies the ideas of the previous two to three names/sefirot: Keter אהיה, Tiferet יהו"ה and Malkhut אדנ"י.
Find the three names, אהיה, יהו"ה ואדנ"י in the sefirotic charts.
We've talked about how we might imagine the levels of Tiferet and Malkhut for the meaning of love. Now, what would be the even more abstract source of love? This would be signified by אהיה, the most transcendent and abstract name of God. Perhaps, for the purpose of our example at least, the אהיה or Keter of love would be the potential that people have to feel anything. This undifferentiated abstract potential would be the context from which the idea of love (parallel to יהו"ה or Tiferet) emerges, and then becomes manifest in concrete examples of love (אדנ"י or Malkhut).
Now that you have the basic structure and the three names, try playing out this logic with other examples. Mathematical truth, for example. If I write down on a page 2+2=4, then I have encountered a concrete manifestation of mathematical truth through the medium of paper, pen and ink. It is here and now. This could be the Malkhut of math. What would the Tiferet and Keter be?
###### 4) His title Adonai is a warehouse that houses all the King's treatures
To make sense of this, consider: All of your concrete loves for particular people at particular times are part of the "set of concrete loves" which is Malkhut. How is this like a treasure house containing all the kinds of love that are included in the abstract idea of love (Tiferet)?
He can you access the different kinds of love that might be found in Tiferet?
###### 5) This being so, know that Hashem has 54 squared names...
We will not yet get into the details of this section, its too much for us at this stage. Just note that the number 54 symbolizes here the many differentiated ways that God is manifest in the world.
###### 6) This being so, the letters of אדנ"י indicated the manner of His governance and dominion.
Pay close attention to what he's doing with the letters here. What does the name אדנ"י mean?
###### 7) Moreover, since no creature in the world...
Here we are applying the logic of Kabbalistic symbolism to Jewish prayer. How does it work? How does this impact on what it means to say that prayer?
###### 8) Now, since Hashem influences all kinds of influence and emanation...
This is actually the beginning of the next section in the text, but it also serves as a beautiful conclusion to the section we've just read. Why is Adonai / Malkut similar to a pool? What does this have to do with blessings?