Hi Cindy,
The law that was up for vote was whether or not people in the 4th largest US city should have anti-discrimination protections. Without them, employers and landlords could refuse to do business with anyone that wasn't protected at the state and federal level (usually just by sex, race and religion). So a trans woman could be fired upon coming out for example, or a gay man could legally be refused housing. The law that was up for vote is pretty standard practice for most US cities in that it extended anti-discrimination protections to 15 new categories of protected classes.
The religious people sized on one tiny aspect of the bill (public accommodation guarantees based on gender identity) and ran a brutally dishonest and harmful campaign against the bill. The proponents of the bill had a squishy response to the most outlandish attacks and of course, got trounced.
So the law doesn't actually forbid trans men and women from using public bathrooms that correspond to their gender identity currently, it just means that those in charge of said bathrooms could refuse to allow access to ladies' rooms to transwoman and vice versa for trans men in the men's rooms, if they feel like making that discrimination.
Hope that helps!