What is the purpose of this axiom in the definition of a uniform space?

The purpose of the axiom is to make sure that this common proof motif from metric spaces will generalize to uniform spaces:

Let $x$ (with certain properties) be given. The enemy chooses an $\varepsilon>0$ and we must then find a $z$ (with certain other properties) within $\varepsilon$ of $x$.

We can do that by first finding an $y$ (with yet other properties) within $\varepsilon/2$ of $x$ and then finding a $z$ with the desired property within $\varepsilon/2$ of $y$. Then, due to the triangle inequality, that $z$ will be within $\varepsilon$ of $x$.

The task is now reduced to proving separately that (1) there is an $y$ near every $x$, and (2) there is a $z$ near every $y$.

In a uniform space, the enemy choosing a $\varepsilon$ corresponds to choosing an $U$, and splitting the $\varepsilon$ into two $\frac\varepsilon2$s corresponds to picking a $V$ by the axiom.

With metric spaces, we sometimes need to prove nearness in a chain of more than two steps, for example splitting $\varepsilon$ into seven $\frac\varepsilon7$s. This would be modeled in a uniform space by several successive uses of the axiom.

Sometimes it is convenient that the pieces you split the $\varepsilon$ into are equal, which is why the axiom gives you a single $V$ such that $V\circ V\subseteq U$, and not just $V$ and $W$ such that $V\circ W\subseteq U$ (which would have been enough to enable the simple case I've sketched above). But the difference is not really important; the $V\circ W$ version implies the $V\circ V$ version because entourages are preserved by intersection.