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In D&D, I am curious if there is any description of the actual mechanics of spellcasting.

After a wizard memorizes a spell, how does he cast it? For example, does he mumble some words to communicate with the world?

Where does the energy come from?

And, take fireball as an example; how does that energy transform into fire elements in a ball shape?

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    \$\begingroup\$ Magic, being entirely fictional, is going to "work" using whatever rules the author decides for the purposes of the game or story. For this reason, the setting - such as Forgotten Realms or Eberron - can make a difference. Did you have a setting in mind? \$\endgroup\$ – PJRZ Sep 16 '18 at 7:51
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    \$\begingroup\$ You may also be interested in this question, answers to which address some of these issues. \$\endgroup\$ – Hey I Can Chan Sep 16 '18 at 8:47
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    \$\begingroup\$ Have you read chapter 10 of the Players Handbook or the Basic Rules? If not, then please do so and refine your question to clarify what it is in the text of the game instructions that you don't understand. \$\endgroup\$ – KorvinStarmast Sep 16 '18 at 12:55
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The D&D 5th edition Player's Handbook, page 201 (under "What Is a Spell"), describes it thusly:

A spell is a discrete magical effect, a single shaping of magical energies that suffuse the multiverse into specific, limited expression. In casting a spell, a character carefully plucks at the invisible strands of raw magic suffusing the world, pins them in place in a particular pattern, sets them vibrating in a specific way, and then releases them to unleash the desired effect—in most cases, all in the span of seconds.

Words and actions

A spell, such as fireball, can have verbal components (words which must be spoken to invoke the magic), somatic components (physical movements which must be made), and material components (objects or substances used to focus the magic).

While material components are frequently detailed, the exact words and movements are not described in the game's rules, and are left to the players' imagination. Forgotten Realms author Ed Greenwood once described how he was asked to include the actual verbal components of spells in a novel, but that the editor ended up deleting them anyway.

In D&D 5th edition, the wizard who wishes to cast fireball must first "prepare" that spell, usually at the start of the adventuring day while well-rested, which fixes the spell in their mind. The Vancian term "memorize" has actually not been used since AD&D 2nd edition. In 5th edition, the wizard casts the prepared spell by performing specific verbal and somatic components.

Those components are never detailed in the rules, but we do know the following about them:

  • They take less than a second.
  • According to Xanathar's Guide to Everything, p. 85, "Identifying a Spell", someone familiar with magic may identify the spell being cast by watching the verbal and somatic components. Someone of that class has advantage on the check. However, success is not automatic even if you know the spell yourself, from which we may deduce that not every spellcaster uses exactly the same verbal and somatic components, and different classes have their own unique verbal and somatic components.
  • Most wizards never invent their own spells. We might conclude that working the strings of magic is a highly complex thing, where learning what earlier wizards have done is still easier than inventing it yourself. We can surmise that it would not be trivial, for example, to make a square or oval fireball, or a lightning-fireball, just by changing one word or gesture. Sorcerers, with a more innate and intuitive knowledge of magic, are more capable of such metamagic.

Source of power

In the Forgotten Realms setting, the source of magic tapped by wizards is termed The Weave, a unique feature of that setting. In other worlds (those of Eberron, Greyhawk, etc), there is no such named source of magic, but spells nonetheless work much as they do in the Realms. Any given DM's setting may have its own unique rules for where magic comes from.

D&D 5th edition describes it simply as shaping magical energies which suffuse the multiverse and drawing energy from them. There's no more specific description given. One might easily conjecture, for example, that fireball comes from the Plane of Fire, but there's no actual source to back that up in the rules, and any number of possible explanations are valid.

In the original D&D, Gygax appears to have based some of his spells on real-world historical practices of folk magic or witchcraft, in which magic often simply works based on specific ritual without necessarily understanding the source of that effect in any modern scientific way. For example, Deuteronomy 18:10, written c. 600 BC, forbids various forms of witchcraft (suggesting that those practices existed at that time), while the concept of a magic circle comes from more recent occult magic (16th century onward).

Ultimately, the way magic "works" in D&D (beyond what is described in the rulebooks) is really up to the player's imagination, and is not described in particular detail.

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You asked several different questions, which I'll answer in general with this quote from the Basic Rules:

What Is a Spell?

A spell is a discrete magical effect, a single shaping of the magical energies that suffuse the multiverse into a specific, limited expression. In casting a spell, a character carefully plucks at the invisible strands of raw magic suffusing the world, pins them in place in a particular pattern, sets them vibrating in a specific way, and then releases them to unleash the desired effect--in most cases, all in the span of seconds.

There are other descriptions of magic in the rules; for instance, there is a section of the Sword Coast Adventurers Guide called "Magic in the Realms" (p. 18-19) which discusses the nature of magic in broad terms of "The Weave". Unfortunately, that doesn't really answer the question of how does a fireball work, and also really only addresses the default setting of Faerun, described in the SCAG.

It might be helpful to think about parallels in real life. You light a match. But how does it work? Well, the friction of the match head on the striker ignites the chemicals in the match head. But how does it work? The point of contact between the striker and the match head get to a very high temperature and causes a chemical transformation that that point. But how does it work? And so on and so forth.

That kind of explanation, down to a set of fundamental set of rules analogous to real-world physics and chemistry is not part of the published rules, and is not generally part of D&D, although it could be, within a specific game. Someone could write a campaign setting in which that sort of mechanics of magic was thoroughly and completely defined.

If you think about great works of fiction featuring magic, is there ever a detailed explanation of the science of magic? So it is with D&D. D&D is kind of a group storytelling for fun, with a set of generally agreed upon rules taken from a published set of rules. People are generally more interested in the story than in creating an imaginary science of magic.

But someone could! It could be you!

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