In matters of debate regarding the existence of God, the philosophy of Gottfried Leibniz, and the writings of predominant philosopher and apologist William Lane Craig, the principle of sufficient reason is an often brought up subject. However, what exactly does this principle mean? Particularly, it can be found in its historical context within the writings of German philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716) in his book On the Ultimate Origins of Things (1697). In it, he writes:
You may indeed suppose the world eternal; but as you suppose only a succession of states, in none of which do you find the sufficient reason, and as even any number of worlds does not in the least help you to account for them, it is evident that the reason must be sought elsewhere. For in eternal things, even though there be no cause, there must be a reason which, for permanent things, is necessarily itself or essence; but for the series of changing things, if it be supposed that they succeed one another from all eternity, this reason is… the prevailing of inclinations which consist not in necessitating reasons, that is to say, reasons of an absolute and metaphysical necessity, the opposite of which involves a contradiction, but in inclining reasons. 
In short, Leibniz is suggesting that a collective set of finite objects requires and explanation of its existence that must necessarily be sought outside of itself. In other words, finite collective sets cannot be the explanation of their own existence. This line of reasoning can be schematized as such:
- (1) Everything that exists, has an explanation of its existence.
- (2) If the universe has an explanation of its existence, that explanation is God.
- (3) The universe exists.
- (4) Therefore, the universe has an explanation of its existence. (from 1, 3)
- (5) Therefore, the explanation of the existence of the universe is God. (from 2, 4).
Premise (1) is what is known as the Principle of Sufficient Reason. Namely (and simply), that any existing thing has an explanation of its existence. As William Lane Craig writes, “This premise is compatible with there being brute facts about the world. What it precludes is that there could exist things which just exist inexplicably. According to (1) there are two kinds of being: necessary beings, which exist of their own nature and so have no external cause of their existence, and contingent beings, whose existence is accounted for by causal factors outside themselves.” 
In light of Craig’s analysis, Leibniz’s principle can otherwise be stated as:
- (1) Anything that exists has an explanation of its existence, either in the necessity of its own nature or in an external cause. 
This line of argument has come to take it’s name most particularly as the Leibnizian Cosmological Argument, which serves as a rather deductive system as opposed to what I think is the more stronger cosmological argument, the Thomistic cosmological argument (an inductive version). Though I will not go in depth to the argument here, I have written in other places about an inductive version of the cosmological argument.
This post was meant to be a straightforward explanation as to what the Principle of Sufficient Reason (PSR) entailed in terms of its function in argument form. Though this principle has taken a mostly modest form over recent years , I personally would withhold usage of this form of cosmological argumentation in light of Kant’s criticism and Aquinas’ strength in his system of the cosmological argument. Nonetheless, many predominant philosophers (see sources below) have made an interesting use of Leibniz’s system of the PSR and made impressive cases for theism out of it.
To finish with Craig’s comment, “Thus, the premises of [the] Leibnizian argument all seem to me to be more plausible than their negations. It therefore follows logically that the explanation for why the universe exists is to be found in God. It seems to me, therefore, that this is a good argument for God’s existence.” 
-  From On the Ultimate Origins of Things, in Leibniz: The Monadology and Other Philosophical Writings, trans. Robert Latta (London: 1898)
-  W. L. Craig, Reasonable Faith, 3rd edn. (Wheaton: 2006) see pp. 106-111
-  Ibid. p. 106
-  see Richard Swinburne, Is There a God? (Oxford: 2010)
-  W. L. Craig, Reasonable Faith, 3rd edn. (Wheaton: 2006) see p. 111