A very brief rundown of some of the arguments Alexander Pruss has offered for accepting the principle of sufficient reason (the principle that for any state of affairs, there is some explanation for its existence).
Epistemological Argument: Once we accept that the PSR is false, and hence some contingent facts have no explanation, then skepticism results: a demon may not be deceiving you, but your perceptions might be happening for no reason at all. Thus if the PSR is false, then we cannot know empirical truths. But we do know empirical truths. Therefore, the PSR is true.
Evolution: The confidence of the theory of evolution is strong argument for the PSR. We presuppose the PSR when examining evolution, otherwise we could conclude that there are some facts with no explanation, and thus that a monkey just appeared out of nowhere and evolved into people.
Inference to the Best Explanation: For a particular phenomenon, we look at all the possible explanations and infer that the best one is the most likely. But we never think to include “no explanation” as a possibility.
No Widespread Violations of the PSR: Why aren’t there violations of the PSR all around us, if it is false? Why aren’t strange things popping into existence all over the place with no explanation? If the PSR is false, then we should think that a brick popping into existence in mid-air is perfectly possible. But we already discount that from having a high probability, so perhaps the PSR is a priori.
Philosophical argumentation: If the PSR is false, then we could just claim brute fact whenever we do philosophical arguments. Why is it true that it is morally right to switch the trolley car to kill one person instead of five? Who cares. It’s just a brute fact.