According to Bereishit Rabbah 18:1, Eve was built out of the rib of Adam:
If Eve was made from Adam's rib, that explains "bone of my bone". But what about "flesh of my flesh"?
JPS Genesis 1: 21And the Lord God caused a deep sleep to fall upon man, and he slept, and He took one of his sides, and He closed the flesh in its place. 22And the Lord God built the side that He had taken from man into a woman, and He brought her to man. 23And man said, "This time, it is bone of my bones and flesh of my flesh. This one shall be called ishah (woman) because this one was taken from ish (man)." 24Therefore, a man shall leave his father and his mother, and cleave to his wife, and they shall become one flesh.
Also, I notice that "bones" is plural. Why might that be?