How does Bertrand Russell’s socks-and-shoes example explain the axiom of choice?

Bertrand Russell’s analogy of socks and shoes illustrates the necessity of the axiom of choice when dealing with infinite collections of sets that lack a defining selection rule.1

• Shoes: Because shoes have a natural, inherent distinction (a left shoe and a right shoe), one can define a simple rule to select an element from each pair—for instance, "always choose the left shoe." Since a clear selection rule exists, the axiom of choice is not required to form a set of one shoe from each pair.1
• Socks: Socks, in this analogy, are indistinguishable from one another within a pair. If you have an infinite number of pairs of socks, there is no inherent property or rule to identify a "left" or "right" sock. Without a rule to distinguish them, you cannot construct a "choice set" by selecting one sock from each pair unless you invoke the axiom of choice, which asserts that such a selection is possible even in the absence of a constructive rule.1

Essentially, the axiom allows mathematicians to bypass the need for an explicit selection algorithm when working with infinite sets.1