Three explanations: (But they all involve that as there are no $x \in \emptyset$ we can not negate $x \le r$ for any $r \in \mathbb R$)
Either $x\in \emptyset \implies x \le r :\forall r \in \mathbb R$ is true or it is false.
If it is false then that implies that there is an $x \in \emptyset$ and an $r \in \mathbb R$ so that $x > r$. That is impossible as there are no $x \in \emptyset$.
If it is true then for every $x \in \emptyset$ then ... something. We can never test this because we can never find an $x \in \emptyset$.
However we can do logic the statement $P \implies Q$ will be true if $P$ and $Q$ are both true, or if $P$ is false. It will only be false if $P$ is true and $Q$ is false. If $P= : x \in \emptyset$ and $Q = : x \le r :\forall r \in \mathbb R$ then $P$ is always false. And $Q$ can never be true for any actuall $x \in \mathbb R$. So $P \implies Q$ is true.
Also a statement is equivalent to the contrapostive. The contrapositive of $x\in \emptyset \implies x \le r:\forall r \in \mathbb R$ is:
$\exists r\in \mathbb R: r < x \implies x\not \in \emptyset$. Well that is certainly true! If $r < x$ then $x$ must exist. And if $x$ exists then $x \not \in \emptyset$!.
If $x\in \emptyset$ then $x$ is a green cheese eating alien who is the reincarnation of Elvis Presley is true because logically a false premise implies all conclusions.
So if $x \in \emptyset$ then $x \le r$ for all $r \in \mathbb R$. (it's also true that $x >r$ fora all $r \in \mathbb R$ and that $x = r$ for all $r \in \mathbb R$ and so on.... as $x$ does not exist we can say anything we want to be true about it.)
Or looking at it another way: If $A_r = (-\infty, r)$ then $\emptyset \subset A_r$ because the empty set is a subset of all sets. (The emptyset has no elements, so no elements aren't in any set.) So if $x \in \emptyset \implies x \in A_r$.
That is true for all $A_r: r\in \mathbb R$. So $x \in (-\infty, r)$ for all $r\in \mathbb R$ so $x \le r$ for all $r \in \mathbb R$.
You might say "there's got to be a trick in there somewhere; nothing can be in all of those intervals" and you'd be right. The trick is that no such $x\in \emptyset$ does exist. But if such an $x$ DID exist, it would have to be in every set including every interval including those intervals.