Why is it so important to "carry" electrons??
A specific answer to this question would depend upon the context - are we talking about atoms carrying electrons, or electrons being carried in an electric current or a wire?
Ultimately the importance of electrons being "carried", or literally moved around, is that electron mobility is the fundamental basis for almost all chemistry, and by extension, the ability to perform work. Chemistry is based almost entirely upon the different properties conferred on elements and compounds by the relationship between their nuclei and their electrons, and so the relative mobility of electrons compared to the nucleus makes it possible for chemistry to actively take place, rather than all possible reactions occurring once, reaching a state of maximum entropy or finding the strongest possible bonds, and then never reacting again. Such a universe would be lifeless and static.
Why electrons, and not another particle, like the proton? Protons are far more massive, for one, and they are subjected to the strong force, which is the reason why nuclei exist in the first place. Electrons are relatively fortunate in that they are inherently far more mobile, while still retaining an EM charge comparable to that of the proton. Why is the electron light, nonresponsive to the strong force, and equal in charge to the nucleus? We don't really know.