OK so the central idea behind magic is, as far as I know, that your mind can change reality directly. Stuff like incantantions are largely about helping your mind focus right. And one philosophical way this can be explained is to say the whole of reality is a projection of our minds, this is why with enough focus the projection, and thus reality can be changed.
Now in Asia certain schools of Buddhism, specifically, the Tibetan Yogacara, or yoga-practicioner school, which is also called the Cittamatra, mind-only school teaches this.
But this would be difficult to believe for a 16th century Christian because without a fixed objective reality, you don't have much use for a creator god, and that would be a nice way to get your ass roasted. Remember, it was kinda hard for Berkeley to explain exactly what role god plays in his system.
There is a book called The Kybalion which claims to convey the principles of Hermetic magic, and it begins with an "all is mind" principle, but apparently it is more based on 19th century philosophical ideas (probably influenced by Asian ideas of the above kind, New Thought, Theosophers, that kind of stuff) rather than actual 16th century stuff.
My question would be, would it be possible for a 16th century or say 13th century European Christian magic practitioner, Kabbalist, alchemist, what have you, to believe "all is mind", and if not, then what was the basis of John Dee's system by what principle did he think magic can actually work?
I can't give you a thorough answer to this, but I can point you toward an excellent one: Al Cummins' The Starry Rubric, which examines how astrology, medicine, ritual magic, and religion all interfaced in the Early Modern period.
One really short answer is that Dee and his angels are part of a long-standing tradition of allegory, of personifying abstract forces... like crediting the workings of luck or fate to Fortuna. From our perspective, it's part superstition and part heuristic (like building a memory mansion to organize one's recall - creating "person-shaped" structures to help process our understanding of and interaction with these hard-to-grapple-with, complex forces).
If you want to get into some primary texts, here's a work by one of Dee's immediate predecessors, the magician (and, let's not forget, natural philosopher - a category that precedes what we today call "scientist") Heinrich Cornelius Agrippa.
Quick sample:
The several Signs, also, of the Zodiac take care of their members: So Aries governs the head and face; Taurus, the neck; Gemini, the arms and shoulders; Cancer, the breast, lungs, stomach and arms; Leo, the heart, stomach, liver and back; Virgo, the bowels and bottom of the stomach; Libra, the kidneys, thighs and buttocks; Scorpius, the secrets; Sagittarius, the thighs and groins; Capricornus, the knees; Aquarius, the legs and shins; Pisces, the feet. And as the triplicities of these Signs answer one the other, and agree in celestials, so also they agree in the members; which is sufficiently manifest by experience, because with the coldness of the feet the belly and breast are affected, which members answer the same triplicity; whence it is, if a medicine be applied to the one it helps the other, as by the warming of the feet the pain of the belly ceaseth. Remember, therefore, this order, and know that things which are under any one of the Planets have a certain particular aspect or inclination to those members that are attributed to that planet, and especially to the Houses and exaltation's thereof. For the rest of the dignities, as those triplicities and marks and face, are of little account in this. Upon this account, therefore, peony, balm, clove-gilly-flowers, citron-peel, sweet-marjoram, cinnamon, saffron, lignum aloes, frankincense, amber, musk, and myrrh help the head and heart, by reason of the Sun and Aries and Leo. So doth ribwort, the herb of Mars, help the head and secrets by reason of Aries and Scorpio; and so of the rest. Also all things under Saturn conduce to sadness and melancholy; those under Jupiter to mirth and honor; those under Mars to boldness, contention and anger; those under the Sun to glory, victory and courage; those under Venus to love, lust and concupiscence; those under Mercury to eloquence; those under the Moon to a common life.
That bit of Agrippa is directly relevant to Cummins' book - it's about how different planets and celestial bodies (for which, read "complexes of emotions" as much or moreso than "physical objects in space" or even "seasons or times in which those objects appear on Earth") affect human bodies... or, really, how bodies and herbs and planets are all part of this interlocking system of correspondences.
The stars are part of a Cosmic Body, and, for example, that celestial ear has a sympathetic link with this herb, which is in turn effective against this human ear ailment.