The significance (from what I understand) is that it allows you to derive things like error estimates and convergence rates based on these spaces. Then when you go to apply these measures to FEA you know it's going to be valid since your basis functions are a part of that space.
You might take...
H(div) and H(curl) refer to function spaces whose div/curl are square integrable in Lebesque sense. Basically function spaces that have continuous function values but whose first derivatives (curl and div operators involve first derivatives) have isolated discontinuities at element boundaries...
