A function by definition is a set of ordered pairs, and also according the Kuratowski, an ordered pair $(x,y)$ is defined to be $$\{\{x\}, \{x,y\}\}.$$
Given $A\neq \varnothing$, and $\varnothing\colon \varnothing \rightarrow A$. I know $\varnothing \subseteq \varnothing \times A$, but still an empty set is not an ordered pair. How do you explain that an empty function is a function?
One thing to point out is that a function is not just any set of ordered pairs; a binary relation is a set of ordered pairs. However, a function $F$ in this case is a special type of binary relation such that for every $x\in\text{dom }F$, there is a unique $y\in\text{ran }F$, such that $xFy$, or $(x,y)\in F$. But $\text{dom }\varnothing=\varnothing=\text{ran }\varnothing$. Can you find some $x\in\text{dom }\varnothing$ such there is not a unique $y\in\text{ran }\varnothing$ for which $x\varnothing y$? Since you cannot, $\varnothing$ is indeed a function.