I can't speak for everyone because I don't know that there is a universal axiomatic understanding, but one way to "understand" finding the root of a given quantity would be that you are peeling off a dimension from a base unit (of, say, area) to arrive at a lower dimensional base unit in the same numbering system.

Another aspect of understanding is _why_ you are doing this, where does it fit into the programme of necessary compression of infinite information density (i.e. the number line is infinitely "dense") so it may be accessible despite the material confines of a human brain and its limited, discretizing capacity for dealing with multiple elements in a single operation. So, philosophically, different lower dimensional spaces integrate to form higher dimensional spaces, in order to facilitate the description of change from one thing into another thing. One linear dimension changing into another linear dimension requires a transition through a quadratic space, from which we get a curve.