trying to restrict abortion rights in Michigan
That's right: sufficient signatures + legislative thumbs-up = automatic passage into law. So you can view the legislature's vote as substituting for the people's vote on an actual ballot measure, or you can view the signatures as replacing the governor's role in the normal legislative process. It's a pretty insane short-circuit of normal procedures, but it's one that conservatives are taking advantage of, in an attempt to force insurance companies to only provide coverage for abortions through separate riders, rather than as a standard part of their regular policies.
That means women would have to be aware of these riders and buy them before ever knowing if they need an abortion. What's more, women would not be able to purchase these riders after becoming pregnant, even in cases of rape. It's outrageous, and even Republican Gov. Rick Snyder isn't on board, which is why activists are trying to go around him. And they're very likely to succeed: They've already turned in far more than that 258,000 signatures they need, and the legislature is all but certain to give its approval to the measure.
For advocates of reproductive freedom, the best remaining avenue of opposition at this point is to try to win back the legislature from the GOP next year. It won't be easy, but Michigan will be a top target for Dems next year. And Republicans are about to give Democrats more fodder.