"State sacrifices" is only a partial thought. Below, I've put brackets around the the complete noun phrase.
I wish I could do fieldwork in the Ming court, observing the court citizens and asking them why [state sacrifices and music] played such a central role in their public and private lives.
So, [state sacrifices] and [music] were two things which were important in the Ming court, and the author would like to ask court citizens why they were so important. It looks like there is a fairly well-regarded book on those topics, so this sentence might refer to that.
"State sacrifices" would be sacrifices performed on behalf of the Ming government. Wikipedia helpfully tells us:
Sacrifice is the offering of food, objects or the lives of animals to a higher purpose, in particular divine beings, as an act of propitiation or worship.