This is a new word that is just starting to gain currency. It means a person who tries to live without plastic. I'm not sure how logical that is as a word - eg, a vegetarian eats vegetables, a fruitarian fruit and so on, so wouldn't you expect a plasticarian to use only plastic? But anyway, this is what it means.
One example of trying to live a plasticarian life is Thomas Smith, a chemistry PhD student, who didn't use plastic for 6 months. He got the idea because, through actually creating plastic materials for his research, he became aware of how long lasting it is (it can outlast a human life) and yet we throw most of it away.
There are health concerns about the amount of plastic we use: some plastics have been linked to serious illnesses. Another major concern is waste - an enormous amount of our waste comes from plastic, and of course it stays in the environment for years.
You can read more about this here.