In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - **Why can't we identify those as the
trivial locales, and what's so great about considering locales
that have no points?**

Anyway, here's the construction of $X$ and $Y$ (taken from [1]). Let $A$ be an uncountable nonempty set (e.g. $\mathbb{R}$) (equipped with the discrete topology), and let $X$ be the set of all functions $\mathbb{N} \to A$, equipped with the Tychonoff topology. For each $a \in A$, let $X_a$ be the subspace $\{f \in X \,|\, a \in im(f)\}$, and let $$ Y = \bigcap_{a\in A} X_{a}.$$ Now the point set $Y_p$ of $Y$ is empty because there is no onto map from $\mathbb{N}$ to $\mathbb{R}$.

In [2, section 5], Johnstone demonstrates why considering such
locales could be useful. The main argument is that *topoi* are
nice things to consider. However, at the point of writing, the
(external) applications of topos theory seem lacking. Hopefully
the situation has changed in mathematics in recent years. Thus
the second question: **How does the consideration of pointless
locales help topos theory, and how does that in turn applies
(externally) to mathematics?**

### Reference

[1] Sketches of an Elephant: A Topos Theory Compendium [Peter T. Johnstone]

[2] The point of pointless topology-[Peter T. Johnstone]