In this chapter, the genealogy continues with the descendants of David and the royal line.
This chapter (in addition to other places) is how we know that Chronicles was written after the exile, because verses 17-24 list some of the royal descendants of king Jehoiachin that all would have lived during and after the exile. In fact, Zerubbabel (mentioned in v. 19) appears in several other books of the bible and he's a central figure in orchestrating Judah's return from the Babylonian exile. So we know for sure that at least some parts of Chronicles are post-exilic.
We also know that many parts of Chronicles are much older than the exile, since a lot of the genealogical material was probably ancient even at the time that Chronicles was written (which is itself ~2500 years old). Most of the historical sections are also largely copied (or at least derived) from Samuel and Kings, but then there are parts like v. 17-24 that are clearly post-exilic, which means that Chronicles must have been compiled, or at least finalized, in the post-exilic timeframe.
I see this chapter as a rushed overview of Judah's history. Through the list of kings, we see Judah's history before the exile, in the golden age when David and Solomon reigned, through to the exile when Jehoiakim had "successors" and not sons (v. 16), through to the post-exilic period when Israel only had governors and no longer kings. Most of these kings form the backbone for the later historical portions, particularly in 2nd Chronicles, so this chapter is almost like a kind of foreshadowing. Nearly all of these kings were already presented in the book of Kings, so my readers should be at least vaguely familiar with them even if you don't remember which are the good ones or the bad ones.
In this chapter the genealogy shifts to a vertical genealogy, giving a chain of descendants, compared to the rest of the genealogy for Judah that is largely horizontal. (see 1 Chron 1 and Gen 10 for previous discussion of the two types of genealogies). Since this vertical genealogy is tightly focused on the descendants of David and the royal line for the southern kingdom Judah, it shows us that this will be an area of emphasis for Chronicles as a whole.
One more minor note I want to put in here is that v. 1 lists "Daniel" as king David's second son. This man is not mentioned anywhere else in the bible, because 2 Samuel 3:3 says that David's second son was named Kileab. Neither Daniel son of David (who should be distinguished from the prophet Daniel, a completely different person) nor Kileab are ever discussed anywhere else in the bible, and given the prominence of David's other sons, this largely suggests that Daniel/Kileab might have died in childhood.