Eowyn fought as
a woman.
She could have trembled in fear before the Witch King, surrendering to hopelessness in full knowledge that there's no way she can defeat him. (After all, she was of the 'race' of Men.) Or she could have chosen to ignore the Witch King's statement and continue to fight as a man, as Dernhelm. In doing so, she would be less likely to be underestimated in terms of strength. A woman in the midst of battle could either have been played around with (so to speak) or slain right away by such a powerful enemy, no matter how skilled she might be.
But Eowyn risked her life by letting down her hair and revealing her womanhood. She most probably had no idea that there is a prophecy concerning her foe's fall, so that knowledge could not aid her. For all she knew, she was just taking a bold yet stupid move, not to mention clarifying things.
But her boldness was rewarded. Where sufficient strength could have been found not, victory emerged.
(I don't know if that was clear, but the general idea sure made sense in my mind.)
*disappears like a bubble*