As a native speaker (Eastern US), I would read this as follows:
As women come to consider being prized by men ...
This sets up the sentence, referring to a changing or developing state of the perception by women of the quality of being prized by men (that is to say the quality of a woman that is her being desired by men).
their ultimate source of strength, worth, achievement and identity,
This, together with the previous quote, indicates that women at some point start to see the quality of being desired by men as the ultimate/underlying/fundamental/most important source of their "strength, worth, achievement and identity". The word "their" is identifying "strength, worth, achievement and identity", as belonging to women.
In summary, the sentence so far indicates that women at some point begin to view all of the "strength, worth, achievement and identity" that belongs to the women as being derived from the fact that the women are prized/desired by men.
Note: I do not at all agree with this sentence, and think it rather sexist.
they are compelled to battle other women for the prize.
This concludes by saying that since women view their worth as coming from the fact that men desire women, the feel the need to fight each other to win the desire of men.