How to create solid blocks of colored background in Apple Pages?
These blocks are simply Shapes that are in the background.  In the template you reference, they are locked.  Click on the "Block" and you will notice little 'x' in the corners.  To unlock so you can move them, select Unlock on the Arrange tab in the inspector.

To create your own shapes, select the insert shape icon on the toolbar.  In the Style tab on the inspector you can change its color, position and other attributes.  See the screenshot below.

Just know that Pages, like many other apps, uses layers.  So move your shapes to the back in order to display text or pictures on top.  You may have several layers as desired for effect.