Wheels
The wheel is a world
In Western esoteric and mystical traditions, the circle represents a at least three classes of things: spiritual, spatial, temporal.
First, perfect things: God, the whole cosmos, the orbits of the planets (pre-Kepler). The trinity is represented in the circle as Father (center), Son (circumference) and Holy Spirit (the radii). In the formulation of Nicholas of Cusa that Bruno would have read, God is a circle of infinite circumference whose center is everywhere. It also represents the soul, as made in the image of God.
Next, circles represent spatial completeness: the world, whether as a globe or as the line of the horizon. Theaters, as in Shakespeare’s Globe, are model worlds. Circles in magical rituals were used to mark off areas of control, containment and separation: boundaries to protect the magician and contain the effects of the ritual.
Third, temporal cycles, ritual time: the clock face, the calendar, the movement of the Sun through the Zodiac. The Prague astrological clock (The Orloj) used as the background of this page, and on the cover of this edition, shows multiple time schemes as well as the positions of the run, moon and Zodiac.
Ramon Llull
Bruno combined the Greco-Roman memory palace technique with the use of lettered and compartmentalized wheels. The use of nested combinatoric wheels derived from Catalan mystic Ramon Lull, who used similar wheels as part of a logical system and art of memory, the Great Art (Ars Magna). Llull’s system combined a list of ten attributes of God, such as Beauty or Goodness, to develop propositions in support of theological or philosophical arguments. The arguments and attributes were demonstrated as existing at all levels of the Great Chain of Being (God, Angels, Humans, Animals, Plants and Matter) in various guises. Llull also used this system as the basis of a theological tool for showing the unity of Christian, Jewish, Muslim, and Pagan thinkers, with an eye toward converting others to Christianity.
Volvelles
Bruno had read a number of Llull’s works, and refers to Llull as one of the line of great memory artists in the prefatory dialogue to De Umbris, but he had an additional source for the kind of circular diagrams he uses here: the paper machines called volvelles. Bruno was many things to many people, but we are certain that he was a professor of astronomy and geometry, a highly popular lecturer at several of Europe’s largest universities at the time. Volvelles were teaching and demonstration apparatuses in common use at the time. Bruno would likely have used paper versions of astrolabes and other astronomical equipment for demonstration purposes during his lectures on Sacrobosco’s On the Sphere, the standard textbook at the time. We know that he was hired on several occasions to deliver lectures on astronomy and mathematics, that he published a commentary on this work, and named On the Shadows of Ideas on yet another (magical) commentary on Sacrobosco by De Cecco.
Cipher Wheels
Leon Battista Alberti published a text, On Ciphers in the mid-15th century, which included the use of a double wheel for encoding substitution ciphers, with the outer wheel fixed and the inner wheel moveable, as well as a system for enciphering a code of predetermined phrases or images.
Bossy’s Giordano Bruno and the Embassy Affair hypothesizes that Bruno worked as a spy while living in the household of the French Ambassador to the court of Queen Elizabeth I of England. While the evidence is somewhat tenuous, the application of Alberti cipher wheels to encipherment, and their existence in Bruno’s work is noteworthy. The memory palace technique itself would be extremely handy for someone carrying out secret missions as well. The possibility that the “magical” material in Bruno’s work is partially a cover for enciphering is also interesting in light of previous discoveries of secret codes embedded in the Steganographia and Polygraphia of Trithemius, which purport to be magical texts.
Za’irja
David Link points out that Llull probably got the idea for his combinatoric wheels from an Arabic divinatory device called a za’irja. This paper device appears to have been in widespread use among traders in the 13th century. What is interesting about the device is that, as with Trithemius’ books, its surface function or “cover” was magical, while the underlying functions may have been related to accounting and mathematical calculation. It is also curious that the za’irja Link references in his paper bear a surprising resemblance to a large diagram that Frances Yates constructed in her book The Art of Memory that combines all of the lists in On the Shadows of Ideas into one massive set of concentric wheels. Again, this seems to be another eyebrow-raising coincidence, as I have no evidence that Yates was aware of za’irja; Bruno never explicitly referenced them, nor did he lay out the concept of the giant Yates wheel in On the Shadows.
De Umbris Idearum 
I’m still having trouble understanding how to use the circles in circles as a memory system.
Think of it as setting a bunch of dials to spell a word. In Bruno’s system, you set a consonant and vowel together using one dial, which is represented by one image. So, if you wanted to memorize the word AMERICA, you could use a set of four dials, set to Aa-ME-RI-CA. In Bruno’s tables in De Umbris, the wheels are set up so that the dials represent 1. a Person, 2. an Action, 3. an Object, and 4. a Circumstance; using the tables of images starting on p.129, those settings give you Rhegima, beguiling, panpipes, treads. So, someone named Rhegima beguiling (charming) someone using panpipes while out for a hike.
Since you’re living in the 21st century, you will probably want to make some adjustments to the system. We don’t use dials or paper wheels very much as a user interface: we just use lists or tables. Bruno’s use of wheels had to do with his interest in the work of Ramon Llull, who built a system of logic and argument around a set of circular diagrams. Both of them wanted to develop encyclopedic, universal knowledge systems, and these simple paper computers helped them to structure their knowledge.
You will also want to construct your own lists of images from popular culture. This is what Bruno was trying to do with his lists; both Ovid’s Metamorphoses and Diogenes Laertes were popular books in his time, so the stories on which the images are based would have been known to his literate readers. Joshua Foer’s book Moonwalking with Einstein does a version of this for his title, choosing a Person: Einstein, performing an Action: Walking, which correspond to whatever it is that he’s trying to remember.
More on the Person-Action-Object system can be found at: http://mt.artofmemory.com/wiki/Person-Action-Object_%28PAO%29_System