What is root(ing) and why do i need this?
To be able to take screenshots, you need access to the lower level graphics layer of android (as there is no real api to do so). 'Rooting' your phone will allow this required access. Some people also refer to this process as 'jailbraking'.

It seems to work on unrooted phones as well.
On most devices access to the lower graphics layer is tightly secured, but some devices come with unprotected access. Depending on your viewpoint, you could consider this as a security hole or a fortunate coincidence.

My phone seems to require root. How do i proceed?
It is an impossible task to follow each root attempt for any combination of phone/android version/carrier. Some phones are quite easy to root, others are more difficult. The solution is quite often only a 'Google query' away.