What is the source of a Paladin's spellcasting ability?

Question

PHB states that:

[...] a paladin's oath is a powerful bond. lt is a source of power that turns a devout warrior into a blessed champion.

and later:

Although many paladins are devoted to gods of good, a paladin's power comes as much from a commitment to justice itself as it does from a god.

So, it seems that conviction, expressed formally by oath, is enough to make a paladin. This is confirmed by answers to the Are paladins required to follow a god? question.

What isn't answered in that question is what is the source of the spells he can cast. I'm OK with paladin powers can come from his stern commitment but spells are clerical ones, chosen and memorized in a way clerics use and definitely more varied and complicated than "Lay on hands" or paladin auras. With all due respect to paladins, I'd say someone intelligent is needed to shape them for a paladin to cast.

So, is it explained somewhere in the books what force grants the ability to cast actual clerical spells to irreligious (or even anti-theistic) paladins?

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Perhaps I'm overcomplicating differentiating between simple paladin powers and true spells but willing magical effects by conviction is a notion fitting Mage: the Ascension, I wouldn't expect it in D&D.

Answer 1

Source of a Paladin's spells: Oath and Belief (PHB, Ch 3 & 10; DMG Ch 1)

On PHB page 205, in the side bar, access to the Weave (from which all magical effects come in D&D 5e) is described as being different for each kind of spell caster.

All magic depends on the Weave, though different kinds of magic access it in a variety of ways. (snip) The spells of clerics, druids, paladins, and rangers are called divine magic. {emphasis in the original}. These spellcasters' access to the Weave is mediated by divine power -- gods, the divine forces of nature, or the sacred weight of a paladin's oath. {emphasis mine}

This information complements the Paladin Class details in Chapter 3 of PHB.

Spellcasting

... you have learned to draw on divine magic through meditation and prayer to cast spells as a cleric does. See chapter 10 ... (PHB, p. 84)

The source is the (sacred weight of the Paladin's) oath, and is bound to a commitment to justice and the cause of righteousness.

The Cause of Righteousness
... a paladin's power comes as much from a commitment to justice itself as it does from a god. (PHB , p. 82)

Divine magic can come from a god, and/or from something more abstract

D&D 5e presents a reality that includes deities who exist without question, but not all divine power comes from those gods. The DMG (p. 11-13) discusses "Other Religious Systems".

Forces and Philosophies
Not all divine powers need to be derived from deities. In some campaigns, believers hold enough conviction in their ideas about the universe that they gain magical power from that conviction. (DMG, p. 13)
(snip) Paladins might serve a philosophy of justice and chivalry rather than a specific deity.

Serving a deity does not preclude serving a Force or Philosophy, and serving a Force or Philosophy does not require a deity. It can be either / or, and it can be both. Even when serving a god, the Paladin's oath and deep conviction (belief) is the conduit that brings them divine power to manipulate the Weave and thus create a magical effect(spell).

The source of the Paladin's power, in terms of the Paladin's spellcasting ability, is in their oath, their deep conviction, and their adherence to it.

That is the source of a paladin's magical powers. (To slightly challenge the frame of your question, it can be more than one thing, rather than being just one thing).

Note: this answer does not contradict the two very good answers by @KRyan and @DavidCoffron

Answer 2

The Oath is the Source of Power

... a paladin's oath is a powerful bond. It is a source of power that turns a devout warrior into a blessed champion.

The Spellcasting feature is part of the abilities that a "blessed champion" receives.

... you have learned to draw on divine magic through meditation and prayer to cast spells as a cleric does.

Meditation and prayer do not necessitate a specific entity, but simply refers to the techniques that strengthen the Oath's power within you. Prayer to a god might be one such way, but praying to a cause or meditating on your resolve is equally powerful to a paladin. Just because they cast some spells that a cleric can also cast does not mean the source of their magic is the same. Almost every caster can cast light, for example.

Answer 3

Dungeons & Dragons has a long history of divine spellpower deriving from abstract concepts. That is, Good, or Evil, or Law, or maybe in this case Justice, are themselves sources of power, that are even implied to have a (limited) will of their own. There are many gods of those things, but those things are, in and of themselves, powerful and willful.

And like gods, that power comes from belief in (the importance of) those things. To the degree that Good is venerated, or Evil is venerated (usually by proxy, say, “power regardless of cost”), there is belief in those things much as there is belief in the gods. And just as gods can delegate that power to their clerics, these concepts can be tapped into by their greatest adherents—in Justice’s case, paladins.

And a paladin adheres to Justice through his Oath. In this way, his Oath becomes his devotion—over and above any god, which he may also venerate, but only so long as that veneration is in line with his oath. In a way, his Oath becomes his “higher calling,” rather than faith in a god, but as far as the magic is concerned, that relationship between paladin and Oath is very much the same as between cleric and god. Which mirrors my answer—from before 5e was even a thing, but nonetheless—about what it really means to be a paladin, saying

Paladins are not beholden to any organization, faith, or even god: they may join with others that they find like-minded, they may worship those deities they think are going to achieve the most Good, but ultimately they answer to Goodness itself. If they discover corruption within their church, or secret evils in their god’s plan, they are beholden to leave that church, forsake that god, and continue to pursue Good.

5e has replaced “Good” with “Justice,” which is perhaps the amalgam concept of Law+Good in this case, but the idea is still the same, as it has been in D&D for quite some time.