I'll try to explain by looking at non-rigid figures.

How to tell if a polygon is RIGID:

If the polygon is made of wooden bars with a hinge at each vertex, none of
the hinges can be turned, IF it is RIGID.

Put a hinge at each corner of a rectangle and you can easily make the
rectangle into a parallelogram.  So a rectangle is NOT RIGID.



Suppose the little circles at the corners are hinges.
Then you could easily transform the rectangle on the left
into the parallelogram on the right, just by pushing on
the left side, or by pulling on the right side.  So a 
rectangle is NOT RIGID. Neither is a parallelogram.  
That's because you can make either change shapes
just by pushing or pulling on a side.

Now look at this triangle with hinges on the corners.
 

Can you change its shape by pushing or pulling on a side? 
NO! You can't make the hinges turn at all! (without bending 
the straight sides). So a triangle is rigid.  

In fact a triangle is the ONLY polygon that is RIGID.

Edwin