They convinced the men (who the majority of legislators were) in the same way that the gay community convinced the 'straights' (who the majority of legislators were) re same-sex marriage.

That said, it's very reductionist to consider the 19th amendment as the moment 'all women' joined 'all men' in having the right to vote. Many women were voting before the 19th (women already had full voting rights in 15 states by the time it was ratified), and many men weren't before, and still weren't for long after, the 19th was ratified.