Multi-layer applies only to pads. To create copper on other layers of a 2 or more layer PCB, you have to either draw each area on the appropriate layer or copy and paste an area from one layer to the other. Why not set up a dummy PCB project and just play with it to explore the possibilities? :)

Note that if you try to set a Solid Region to Multilayer, EasyEDA will automatically reconfigure the Solid Region to Board Cutout which will create a hole in the board.