Because of Goedel's Incompleteness Theorems, we know that
we cannot describe a complete axiomatization of
mathematics. Any proposed axiomatization $T$, if
consistent, will be unable to prove the principle Con(T)
asserting that $T$ itself is consistent, although we have
reason to desire this principle once we have committed
ourselves to $T$. Adding the consistency principle Con(T)
simply puts off the question to Con(T+Con(T)), and so on,
in a process that proceeds into the transfinite.

Thus, we come to know that there should be a transfinite
tower of theories above our favorite theories, transcending
them in consistency strength. The incompleteness theorems
imply that there is a tower of theories above PA, above
ZFC, each level transcending the consistency strength of
the prior levels.

How fortunate and wonderful that we have also independently
come upon such a tower of theories: the large cardinal
hierarchy. Numerous large cardinal concepts arose very
early in set theory, from the time of Cantor, before
Goedel's theorems and before the notion of consistency
strength was formulated. These large cardinal concepts
arose from natural set-theoretic questions in infinite
combinatorics: Can there be a regular limit cardinal? Can
there be a countably-complete measure measuring all subsets
of a set? Does every $\kappa$-complete filter on a set
extend to a $\kappa$-complete ultrafilter? And so on.

Eventually, it was realized that these large cardinal
notions separate into a very tall hierarchy, with the
property that from the larger cardinals, one can prove the
consistency of the smaller cardinals. For example, if
$\kappa$ is the least Mahlo cardinal, then the universe
$H_\kappa$ is a model of ZFC + *there is a stationary
proper class of inaccessible cardinals* + *there are no
Mahlo cardinals*. If $\delta$ is the least measurable
cardinal, then $H_\delta$ satisfies ZFC + *there are a
proper class of Ramsey cardinals, but no measurable
cardinal*.

Thus, the large cardinal hierarchy provides exactly the
tower of theories, whose levels transcend consistency
strength, that we knew should exist. And it does so in a
way that is mathematically robust and interesting, with its
foundations arising, not in some syntactic diagonalization,
but in mathematically fulfilling and meaningful questions
in infinite combinatorics.

The case of Vopenka's principle is just like this. VP is a
large cardinal axiom at the higher end of the large
cardinal hierarchy, implying the consistency of the
existence of supercompact cardinals, say, which are far
stronger than strong cardinals, which imply entire towers
of measurable cardinals, which imply numerous Ramsey
cardinals and so on down the line.

Illustrating the essential large cardinal nature, the VP
axiom is elegantly stated: for every proper class sequence
$\langle M_\alpha | \alpha\in\text{ORD}\rangle$ of first
order structures, there is a pair of ordinals
$\alpha\lt\beta$ for which $M_\alpha$ embeds elementarily
into $M_\beta$. (It is equivalently stated in terms just of
graphs, if you like.) It's simple and clear---beautiful!
And the consequences are far-reaching and often profound,
as you have observed in category theory, in the way that VP
implies that the set-theoretic universe is regular and
organized.

These are the reasons you should be attracted to Vopenka's
principle. It is an elegant combinatorial principle, with
far-reaching consequences that interest you, which has not
yet been refuted.

In contrast, I find the philosophical heuristics that seek
to justify the large cardinal axioms, on the grounds of
reflection or some other means, to be ~~so much hot air~~ ultimately unsatisfying.
These arguments are not mathematically sound, and cannot be
made to be, by the Incompleteness Theorems.
Philosophically, they seem much more like rationalizations
after the fact. For example, even at the much lower (and
therefore seemingly easier-to-justify) level of
inaccessible cardinals, one sometimes hears an appeal to
reflection type views, that since we have no definable
unbounded map from a set into the ordinals, that there
should be a level $V_\kappa$ of the universe also with this
feature, and that such a level would be inaccessible
cardinal. Of course, the conclusion outstrips the argument,
with the conclusion seeming to justify at most
$V_\kappa\models$ZFC, which is a weaker notion, and the
meta-reflection principle appealed to amounts anyway to a
large cardinal principle of its own.

Ultimately, we must recognize the uncertain nature of all
our mathematical enterprise. As our hypotheses rise higher
in the large cardinal hierarchy, we must become less sure
of consistency---perhaps they will be shown to be
inconsistent. This issue arises even at the lowest levels
of our mathematical axiomatizations, for we may find at any
time (as mentioned in a recent MO
question)
that even PA is inconsistent. As Woodin says, we all have
in our minds the image of a railway line, lined by a
sequence of telegraph poles, proceeding into infinity; but
when the physicists tell us that the universe is finite, we
realize that this picture is pure imagination. Perhaps it
is simply inconsistent? So skepticism about consistency has
nothing especially to do with the infinite.

Meanwhile, the large cardinal axioms are fascinating and
have fascinating consequences. Let's seek out the boundary
of consistency, with an attitude tempered by the
realization that we may find inconsistency.

In summary, we cannot ever be sure that our axioms are
consistent, and we know that above the mathematical theory
about which we may be sure, there is a tall tower of
theories whose levels transcend consistency. Among them are
fascinating theories that are elegantly stated with
far-reaching consequences, and which we have not yet
refuted. So let's study them! Let's find the boundary
between consistency and inconsistency!