Special Relativity is special because it places an upper bound on speed: speeds faster than that of light are simply impossible. What does this really amount to, though? - an upper bound on the speed of time. Why? Well, if we distinguish different moments in time by the different events which constitute them, and if every event has a cause, then temporal change is constrained to the same speed as causal change, i.e. the speed of light.
So we've constrained time, but what about space? Doubly Special Relativity posits a limit on space analogous to the limit on time. What does this amount to? - a smallest unit of length. Just as there is a fastest speed, there is a smallest size.
Of course, this smallest size would be pretty damn small. However, the positing of a smallest size resolves some ancient conceptual problems about infinity. If space is a continuum, i.e. infinitely divisible, i.e. there is no smallest unit of length, then we seem to have a real infinity right under our very noses. Zeno's paradoxes all turn upon the puzzling nature of such an infinite. Of course, resolution of paradox alone isn't a good enough reason to adopt a theory, but it does increase its intuitive interest. . . .