Not all countries will follow, because some countries do not care about international law. But there are many counties that will folow this new law, and hopefully it will help reduce some of the detrimental effects that humans have had on the enviroment. Hopefully the law will result in some major changes.
With the Paris mate change agreement becoming international law, I do believe that other countries will follow it and enact climate change laws of their own. The environment and issues of climate change have gained a lot of momentum in recent years, as the threat of pollution and environmental destruction are at an all time high. Countries need to look at agreements, such as the Paris climate agreement, and come up with solutions of their own.
With the agreement becoming international law, countries will not have a choice in whether to follow it. Countries that try not to follow it will have to face harsh sanctions. The international community could use it as an excuse to treat these countries poorly. The world will fall in line now that it's the law.
If you are a decent human being than you have to care about the future of your children and the only planet we have. We all live here. We have to keep our planet safe. 99% of all scientists across the globe agree that this is the single largest threat to human existence. I think people will take it seriously.