Sword and Sorcery (5e)
I It is easiest to envision The Tempest as a great ocean which permeates the multiverse. Eddies and Currents flow forth from the very land. Each Celestial sphere exerts a gravitational like influence over these currents of magic. This creates tides in which different magical influences fluctuating at any given time.
The first thing most trained spellcasters learn is how to perceive these flows, currents, and eddies.
After years of study a wizard has memorized planar charts and learned the exact way in which you vary your spell formulas to create consistent effects with in this ever changing landscape of energy.
The cleric has spent years learning celestial hierarchies to better interpret which divine agent will most likely hear and respond to their supplications.
The sorcerer has an innate understanding of the flows currents and eddies which allows them to interact with them more easily.
Warlocks are taught extremely specific spell forms by their patrons. These often archaic spells find ways to bypass the current fluctuations but are extremely exhausting/costly to perform.
As with all other things in the multiverse sound and light interact with magic. Crystals are often used as focuses for spell casting but the vibrations produced by musical instruments are easily shaped and give the caster fine control. Bards have an ear for magic in much the same way as sorcerers have it in their blood.
Different materials have different properties. Any mountain gnome can tell you this. Spellcasters understand the metaphysical properties of items. They use them to harness or collect the proper energies to enact their spells.
Each movement a spellcaster makes is a physical extension of their desire. These movements are used to parse, slice and regather the invisible but tangible magic around them.
The vibrations created by intoning sounds acts in the same way as all the above components. Sounds are strung into syllables are strung into words. This is why the language of magic is so varied and why the words have no literal translation. Also, this is why the language of the dragons is so apt for spell casting, it has a more nuanced tonal range.
As a spell caster progresses in the art their perceptions and learning expand. This allows them to better perceive the currents around them and interpret how best to harness them.