If I understand it correctly, the Chinese dynasties start with the Xia-Shang-Zhou dynasties. However, writing starts during the Shang dynasty with oracle bones, so how do we know that the Xia dynasty really existed ? How do we know what happened ? It seems to me that the timespan between the supposed-Xia and the later historians (~800 years) is too long and that their writings doesn't hold any historical value.
So, how do we know about what happened during the Xia Dynasty in Ancient China if there was no writing system yet at that time ?
The problem of the existence of the Xia Dynasty relies precisely upon the inability to tie written political history to discovered archaeological evidence.
The Shang Dynasty's existence was confirmed because artifacts dug up (like oracle bones) dated to that era had written inscriptions on them, that when deciphered as the names of Shang kings, matched up with the order and names of Shang kings listed in Sima Qian's history.
Because some of the proposed cultures like the Erlitou culture for the Xia dynasty are pre-historical (meaning had no writing or no writing has been deciphered), there's no way to actually match up the known written record to the archaeological record.
Which means your question, "how do we know about what happened during the Xia Dynasty in Ancient China if there was no writing system yet at that time" is dead on the mark, and all such claims as to the corroborated existence of the Xia dynasty are questionable and likely political (meaning there's a political interest in legitimizing the length and continuity of Chinese civilization).
And as for /u/skyanvil's claim of oral passage of history, much anthropological research has shed light on the unreliability of oral history when it extends past living memory (usually about 2 generations) because the oral history tends to be more for explaining how the world came about according to the best understanding currently, rather than an accurate recitation of past accounts.
Which is why written history tends to be preferred. Though it has the same "explain the world as best understood currently" problem, it is at least a "frozen-in-time" account of that thought pattern for that particular time period, rather than one continuously polished into the present.
Source:
Lindqvist, Cecilia. China, Empire of Living Symbols. Reading, MA: Addison-Wesley, 1991.
Keay, John. China. London: Harper, 2009.
Not to dissuade anyone from answering you directly, but a similar question was asked a few hours ago. The main answer there might have what you're looking for, or you could ask some follow-up questions if you need more.
They are finding new evidence for the Xia dynasty all the time.
for example: http://www.nzweek.com/technology/chinese-archaeologists-uncover-4000-year-old-fortifications-109117/
The Shimao ruins is dated to the time of Xia Dynasty, and establishes that the Chinese were building very sizable city states at the time.
You have to understand that even during the Shang dynasty, much of Chinese history was ORALLY passed down, not written, but still based upon written history. It's just that we haven't found all the written stuff from that long time of ago yet.
But like I said, they are finding new evidence for Xia and Shang all the time now.