What caused the earth's climate to stabilise ~10 thousand years ago?
You need to examine the assumption that it "stabilized". Rather (and simplistically, of course), the Milankovitch cycle en.wikipedia.org/wiki/Milankovitch_cycles moved from a stable Ice Age regime to an interglacial one.  Absent the current human interference, it would (IIRC, anyway) continue the interglacial for some tens of thousands of years, before entering another ice age.

Nor does the end of the last Ice Age really correlate with the rise of large civilizations, since there was about a 5-6K year gap.  IOW, roughly as long from the end of the Ice Age to the first civilizations, as from them to the present.