Difference between order of convergence and order of consistency?
A quick and dirty answer: by definition.

Let's use the definition from Wikipedia.
Let $\delta^h_{n+k}$ be the local error.
A method has order $\textbf{p}$ if
$$\delta^h_{n+k} = O(h^{\textbf{p+1}}) \text{ as } h \to 0.$$

Notice $p$ and $p+1$ in the definition. Since the explicit Euler method has local error of order $O(h^2)$, $p+1 = 2$, hence $p = 1$.

On the other side, if you want to know the reason behind such definition, then I think it is about showing you the part of the solution's Taylor polynomial which the method reproduces correctly (by giving you the degree of the last correct term).