[1] Zeno's paradoxes state that if one were to examine individual moments of an object moving in a given direction, (as with an arrow flying towards a target) or overtaking a second, slower object, (as with Achilles challenging a tortoise to a race) one would not actually be able to find a moment of the act of change or motion taking place, thus proving Zeno's view of motion as illusory and impossible.
The literal solvitur ambulando originates with Diogenes the Cynic in an account by Simplicius of Cilicia of a debate against Zeno; upon hearing his opponent's argument, Diogenes silently rose and walked away, thus implying the reality of motion to be so self-evident that any attempt to debate it is meaningless.
Later assertions of a superficially obvious reality in the face of abstract philosophical quandaries include Dr Samuel Johnson's appeal to the stone and Ludwig Wittgenstein's "hinge" epistemology.
Chatwin, who "passionately believed that walking constituted the sovereign remedy for every mental travail", learned it from Patrick Leigh Fermor and immediately wrote it down in his notebook.
[3] In this sense the phrase can be used as a truism for the wider physical and mental benefits of walking and other forms of regular exercise.