In this chapter, the historical narrative begins with the death of Saul.
This is a relatively short chapter, but it's important because it reveals a lot about the nature of the historical narrative and its connection to the genealogy that we just read.
So. First off, my readers may notice that this chapter is almost exactly copied from 1 Samuel 31. This is important for a couple reasons. First, it shows that the author of Chronicles is familiar with the book of Samuel, since this is almost a word-for-word copy. Second, the author of Chronicles is skipping over an enormous volume of the story from 1 Samuel 1-30, which includes things like: the birth of Samuel, the selection of Saul as king, Saul's sins, the selection of David as king, and the lengthy conflict between David and Saul. None of these things are discussed in Chronicles, but because the story begins here with the death of Saul, the author of Chronicles assumes his readers are already familiar with Saul, David and the war against the Philistines. Since we know that Chronicles was written hundreds of years after these events, it is likely that the book of Samuel must have been relatively widespread in the time of the Chronicler. At least, it is just as widespread as the Chronicler intended his own book to become.
We see this most clearly in verses 11-12, where the Chronicler tells us how the men of Jabesh-Gilead honor Saul, but without telling us why. He simply assumes that we know about how Saul delivered Jabesh-Gilead from the Ammonites in 1 Samuel 11. In 1 Samuel 31, this story would have made complete sense because the deliverance of Jabesh-Gilead happened earlier in the book; in Chronicles, it only makes sense if we have also read 1 Samuel.
Actually, this is the reason why verses 13-14 are included. These verses do not exist in 1 Samuel, they are only present here, precisely because Chronicles does not have the prior narrative about the life of Saul with which to explain his death. In 1 Samuel 28, it is very clear that Saul is going to die in battle as punishment for his sin and disobedience against the LORD; no further explanation is required in 1 Samuel 31 when he actually does die. In Chronicles on the other hand, the story begins with Saul's death, and the Chronicler felt that it was appropriate to give a (highly abbreviated) explanation. These two verses still assume that the readers know the story from 1 Samuel because they don't bother to explain the situations in which Saul sinned.
Lastly, I'm not sure why the author of Chronicles began his story here. Interestingly, this is the same point where 1 Samuel and 2 Samuel are divided, but keep in mind that Samuel was a unified book until the time of the Septuagint (~70 BCE), which is actually older than Chronicles. Samuel was a single book at the time that Chronicles was written, so if anything it's possible that Samuel was divided here because this is where the Chronicles narrative begins. I think more likely than that, it is just a natural breakpoint in the story because this is the transition from the "evil king" Saul to the good king David. Given the context of Chronicles, how it was written when Judah was returning from exile, it's probable that the Chronicler is trying to be more hopeful and inspiring to his readers and wishes to not dwell too long on Israel's troubled past and focus more on the glory days of David and Solomon.
In the next chapter, that glory begins in earnest with David crowned king over all Israel.