Why is the idea that two points on a rigid body always correspond to the same distance so important in special relativity? In Einstein's chapter on physical meaning of geometrical propositions (The special and general theory of relativity) he wrote about supplementing Euclidean geometry with the idea that two points on a rigid body always correspond to the same distance.

If, in pursuance of our habit of thought, we now
  supplement the propositions of Euclidean geometry
  by the single proposition that two points on a
  practically rigid body always correspond to the
  same distance (line-interval), independently of
  any changes in position to which we may subject
  the body, the propositions of Euclidean geometry
  then resolve themselves into propositions on the
  possible relative position of practically rigid bodies.

Besides, I got the feeling that such a supplementation is important in the scope of special relativity.
But I do not see why he insisted on this supplementation. Special relativity does not deal with curved spacetime so it should be straightforward to see it. Or I may be missing something here...
 A: 
Why is the idea that two points on a rigid body always correspond to the same distance so important in special relativity?

I don't think it's so important in special relativity, just in this particular presentation of SR. Most presentations of SR never talk about rigid bodies, which are a somewhat problematic thing in relativity.
What Einstein is trying to get at is that when we measure things, we can only measure them using measuring instruments.
Rather than Einstein's rigid bodies, one can instead use rays of light and a single clock. A nice presentation in this style is Geroch's General relativity from A to B.

Special relativity does not deal with curved spacetime so it should be straightforward to see it.

Actually if we tried to apply this to general relativity, it wouldn't work. In Riemannian geometry, we can't in general transport a rigid shape from one place to another, i.e., congruence doesn't exist in general.
No matter what set of measuring implements you use, you need at least one that defines some unit of measure. If you have a clock, then you have a unit of time measure (the second).
In Euclidean geometry, the way we implement measurement is by arbitrarily picking some line segment and using it as our unit. We can compare it to other things because objects can be transported and compared for congruence.
