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?
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.
Best Answer
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.
This information complements the Paladin Class details in Chapter 3 of PHB.
Spellcasting
The source is the (sacred weight of the Paladin's) oath, and is bound to a commitment to justice and the cause of righteousness.
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".
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