One historical anecdote can begin to provide an answer to this question. Elizabeth Cady Stanton and her husband, Henry Stanton, were both abolitionists. They attended the World Anti-Slavery Convention in London in 1840 on their honeymoon. Once there, however, Elizabeth Cady Stanton, along with Lucretia Mott, who became her lifelong friend and colleague, was barred from taking a seat at the convention. This affront was a seminal event in Stanton's life; after it, she committed her life to fighting for women's rights.
This incident illustrates a broader trend in antebellum America. Women, especially middle-class Northern women, became involved in a number of reform movements, including abolition. This proved an important political outlet for women who were otherwise excluded from the political process, and for many (but not all), it served as a springboard for broader political activism. Aside from snubs like the one experienced by Stanton and Mott, the egalitarian ideology espoused by abolitionists naturally extended to gender relations as well. Women were active and respected figures in the abolitionist movement, and many male abolitionists, including William Lloyd Garrison and Frederick Douglass, supported women's rights. In short, attacks on the injustice of slavery not only led many women to question the injustices that affected them, but empowered them to speak up against them. Stanton acknowledged this connection when she spoke to a prominent anti-slavery society:
[T]his is the only organization on God’s footstool where the humanity of woman is recognized, and these are the only men who have ever echoed back her cries for justice and equality….the mission of the Radical Anti-Slavery Movement is not to the African slave alone, but to the slaves of custom, creed and sex, as well...