From Latinfamilia(“family”). The old sense of "family" in Romanian is now mostly lost, replaced by the neologism familie. Compare Aromanian fumealji, which retained the sense of "family". Despite the phonetic and semantic similarites, is unrelated to Latin femella, and it is improbable that this word influenced the development of the meaning in modern Romanian because the older forms of femeie did not have the sense of woman and the other Romanian/Eastern Romance languages don't have this sense either. It is also unrelated to Latin femina, which produced some apparent "cognates" in the Western Romance languages that only share superficial similarities.
For a similar semantic development, see Albanian fëmijë.