Let $A$ and $B$ be sets. In our case, $A=\{0,1\}$ and $B=\{1\}$. But let us be general for a while.
A relation from $A$ to $B$ is any subset of the set of all ordered pairs $(a,b)$, where $a\in A$ and $b\in B$. In our case, there are only $2$ such ordered pairs, namely $(0,1)$ and $(1,1)$. So the set of ordered pairs is the set $\{(0,1),(1,1)\}$.
This is a two-element set. So it has $2^2$ subsets. We can list them all explicitly: they are $\emptyset,\quad \{(1,0)\},\quad \{(1,1)\}, \quad \{(0,1),(1,1)\}.$
Which of these $4$ sets of ordered pairs are functions from $A$ to $B$? Formally, a function from $A$ to $B$ is a set $F$ of ordered pairs $(a,b)$ such that for any $a\in A$, there is a unique $b\in B$ such that $(a,b)\in F$.
Here there is no uniqueness issue, since $B$ has only one element. But for every $a$ in $A$, we must have a $b$ such that $(a,b)\in F$. So that must hold for $a=0$, and also for $a=1$. The only set of ordered pairs that qualifies is $\{(0,1),(1,1)\}$: one set of ordered pairs, out of the $4$ sets of ordered pairs available.