EDIT: As pointed by Stefan H. in his comment, the solution I have suggested only works if $X$ is normal, since I am using Tietze extension theorem.
Perhaps I have overlooked something and I will be blushing, but I will give it a try. (This is my solution, I did not check the books I mentioned in the above comment. Perhaps the proofs from those books can give you a hint for a different proof.)
Let $x_n\in X$ be a sequence which converges to $x\in\beta X\setminus X$. We can assume that $x_n$'s are distinct.
I will show bellow that $\{x_n; n\in\mathbb N\}$ is closed discrete subspace of $X$. But first I will show how to use this fact.
For any choice of $y_n\in[0,1]$, $n\in\mathbb N$, we can define $f(x_n)=y_n$ and extend it continuously (by Tietze's theorem) to the whole $X$.
Now there exists a continuous extension $\overline f : \beta X \to [0,1]$.
By continuity, the sequence $y_n=\overline f(x_n)$ converges to $\overline f(x)$. We have shown that every sequence in $[0,1]$ is convergent, a contradiction.
Now to the proof that $\{x_n; n\in\mathbb N\}$ is closed and discrete.
Since $\{x_n; n\in\mathbb N\}\cup\{x\}$ is a compact subset of $\beta X$, it is closed in $\beta X$.
The intersection with $X$ is $\{x_n; n\in\mathbb N\}$ and it must be closed in $X$.
Now we consider $\{x_n; n\in\mathbb N\}$ as a subspace of $X$ and we want show that this subspace is discrete.
Choose some $x_n$. By Hausdorffness, it can be separated from $x$, i.e. there exists a neighborhood $U\ni x$ such that $x_n\notin U$ and a
neighborhood $V\ni x_n$ with $V\cap U=\emptyset$.
Now by convergence $U$ contains all but finitely many $x_n$'s, hence using Hausdorfness we can separate $x_n$ from the (finitely many) remaining ones.
In order to show that $\beta(\mathbb Z)$ is not first countable, it will suffice to show that $|\beta(\mathbb Z)|\gt2^{\aleph_0},$ since a Hausdorff space which is separable and first countable has cardinality at most $2^{\aleph_0}$ (each point is the limit of a convergent sequence of points in a countable dense set).
The space $C=\{0,1\}^\mathbb R$ is a separable compact Hausdorff space. Define a countable dense subset $S\subseteq C$ and a surjection $f:\mathbb Z\to S.$ Since $\mathbb Z$ is discrete, $f$ is continuous, and therefore extends to a continuous surjection $g:\beta(\mathbb Z)\to C,$ showing that $|\beta(\mathbb Z)|\ge|C|=2^{2^{\aleph_0}}\gt2^{\aleph_0}.$
Best Answer
Even for spatial locales this is not true, since there are (sober) topological spaces whose maps to their Stone-Čech compactifications are not injective. An example is the line with two origins.
I don't know a snappy characterization of those locales for which the map $r$ is a monomorphism, and I'm not sure you'll be able to find one, because monomorphisms are not viewed as the "right" notion in the category of locales. Much more attention gets paid to regular monomorphisms (e.g. a sublocale of $L$ is defined to be an isomorphism class of regular monomorphisms into $L$).
And the theorem is that $r$ is a a regular monomorphism if and only if $L$ is a completely regular locale. Similarly, the map from a topological space to its Stone-Čech compactification is a topological embedding if and only if the space is completely regular.
The best reference for this (and all related topics) is probably Johnstone's book Stone Spaces.