Why should anyone care about what figures can be constructed using only these two specialized tools? Why not choose different tools, or more tools?
One reason for the choice is that the fundamental objects in classical plane Euclidean geometry are points, lines, and circles, and a straightedge and compass are the minimal set of tools that will allow any circle or any line to be drawn. So it is obviously interesting to look at what can be drawn (constructed) using only the two tools that are obviously necessary to draw all the sorts of figures that are interesting.
The other reason for the choice of a straightedge and compass is that this minimal set of tools enables one to make an amazingly large collection of constructions. Also, for thousands of years, many people thought that any ``reasonable'' construction could be done with just those two tools. There were, however, 3 famous constructions (see Section 6) that nobody seemed to be able to do (using just a straightedge and compass, that is).
Finally, doing classical straightedge and compass constructions provides a wonderful selection of problems that helps to reinforce important geometric concepts from all parts of geometry.