Near as I can tell, the basic point is that gender based discrimination and prejudice needs to be applied across the board.
Feminism focuses on the needs of women, and works to bringing women to an equal footing in all aspects of society. By nature of that effort, that doesn't include men's issues since social and cultural norms already have men at the standard of what we're all supposed to have.
However, there are still plenty of issues where prejudice, stereotypes, and harmful norms affect men. Anti-sexism serves to also work against those, as well as the issues of anyone that doesn't match the man/woman binary.
So, feminism would be an important aspect of anti-sexism, but not the entirety.
See, feminism does result in outcomes that benefit men as well, but that's a side benefit rather than the goal. With anti-sexism, it is a goal as important as any other. Since there's a synergistic effect, I might even argue that anti-sexism efforts boost feminism in a way that it can't reach on its own just by virtue of not having to be held back by slower progress in men experiencing the freedom that feminist thinking gives.