By your tags you've asked for a soft answer, and so let me
try to provide one.

The theorem is indeed very nice and engages with and
reinforces a number of philosophical views in set theory.

First, there is the idea that large cardinal axioms are
leading us towards the final, true set-theory, and so
set-theorists are keenly interested when the existence of
large cardinals causes a fragment of set-theoretic truth to
stabilize in the lower part of the universe. One could take
this to mean that that stabilized fragment is a part of the
final theory we seek. Although Gödel's hopes that
large cardinals would settle CH were dashed by the
Levy-Solovay theorem, nevertheless the phenomenon does
exist. Increasingly large large cardinal assumptions, for
example, imply that increasingly large portions of the
projective hierarchy have extremely nice properties
(Lebesgue measurable, property of Baire, determinacy,
etc.). If there are infinitely many strong cardinals, then
it is consistent that projective truth is invariant by
forcing.
When there is a proper class of Woodin cardinals, then the
theory of $L(\mathbb{R})$ is invariant by set forcing.
This paper by Neeman and
Zapletal
shows that if there is a weakly compact Woodin cardinal,
then for any proper forcing extension $V[G]$, there is an
elementary embedding $j:L(\mathbb{R})^V\to
L(\mathbb{R})^{V[G]}$. The idea is that once we have
sufficient large cardinals, then one cannot change the
universe too much by this kind of forcing.

Viale's theorem essentially extends this from
$L(\mathbb{R})$ to $L(P(\omega_1))$, which is impressive.

Second is the philosophical idea that much of the
indeterminism of set-theory is due to the anomalous effects
of forcing. That is, the extreme flexibility of forcing
causes so much set-theoretic chaos---we can turn the
continuum hypothesis on and off like a lightswitch---and so
when we come to know that a statement or class of
statements is invariant by forcing, or by a huge natural
class of forcing such as proper forcing, then this is
really significant. The theorem and the others I have
mentioned are all instances where the chaotic nature of
forcing is controlled or restricted by the existence of
large cardinals.

Third, there is the view of Martin's Maximum MM as
expressing a fundamentally important truth about the
universe. Justin Moore has emphasized its attractive nature
as a fundamental axiom generalizing the Baire category
theorem. Viale's theorem essentially says that when there
are sufficient large cardinals, then Martin's maximum
completely smoothes out the chaotic effects of (proper)
forcing, since under statements in $L(P(\omega_1))$, which
is a huge fragment of the universe, cannot be affected by
proper forcing preserving MM. This result therefore
underscores the idea that MM and large cardinals cause a
measure of stability in the set-theoretic universe. (Note
that the theorem is definitely false without the MM
preservation, since proper forcing can switch the CH
lightswitch, and CH is expressible in $L(P(\omega_1))$.)

Meanwhile, I expect that other set-theorists can add
important insights.