Here's how we know, in under 1000 words! A five-minute read.
For this discussion to be applicable to as much as possible, we will generalize the notion of a description. These "generalized descriptions" are not descriptions in the colloquial sense of "a spoken or written representation", but in the sense of a thing that in any way exists or a thing that is in any way known.
We don't need to bother with the ontological ("existing") or epistemological ("knowing") statements or assumptions about a generalized description: if anything is outside the generalized description, it is outside the colloquial, spoken-or-written notion of a description, and outside any notion of existence, and outside any notion of knowledge, entirely. Even when we describe something which is partially outside these notions, it's not entirely outside of them because we have described it! Something that is not a generalized description would have to be entirely colloquially indescribable and entirely non-existant and entirely not-notknowable.
Anything we describe, in the colloquial sense, is a "description", in the generalized sense. A flower; a fictional flower; the idea of a flower; all flowers; one specific red flower; honesty; beauty; love... everything is a description.
Now that everything is a description, we state: a description which is identical to a definition is metaphysical. But what's "metaphysical", and what's "identical", and what's "a definition"? The principle of contradction will sort these out!
The principle of contradiction is the mutual exclusion of descriptions. "Mutual exclusion" in this sense is for one description or the other or both to be excluded. Excluded from what? Excluded from a description that adds together other descriptions.
Since everything is a description, several descriptions added together as a whole are a description. Within this added-together description, one description may be described to exclude another. When parts of a whole exclude one another, the whole excludes itself: the whole is not whole without the excluded part. Thus we have the principle of identity from the principle of exclusion: a description that does not exclude itself is identical to itself.
A description that does not exclude itself, and will not have other descriptions added to it or taken away from it as parts of itself as a whole, is a "definition". Our meaning of definition is an immutable not-self-excluding description.
We described what's "identical" and what's "a definition". But why are definitons metaphysical, while other descriptions may not be metaphysical? And what's "metaphysical"?
What is metaphysics? Our meaning of metaphysics is any collection of definitions: a collection of immutable descriptions that follows the principle of contradiction.
Different metaphysics may contradict each other; for example, in Euclidean geometry the sum of the angles of a triangle is always equal to the sum of two right angles, but in non-Euclidean geometry this is not always true. These two metaphyhsics, Euclidean geometry and non-Euclidean geometry, exclude one another: "this is always true" excludes "this is not always true". However, each one within itself does not necessarily have any definition or definitions that exclude any other definition or definitions. Each one of them could still be, individualy, our type of metaphysics, although when taken together they are not.
Other descriptions of metaphysics may not follow the principle of contradiction; they are not metaphysics in the sense used here; they are indiscriminate, and within them "anything goes": there is no principle that sets apart what is or isn't excluded.
For our metaphysics, there are descriptions called algebra. Algebra is a metaphysics of symbols and rules for those symbols. Algebra is what describes math and logic. When a metaphysics is described with algebra, it is said to be "formal". We will find that some formal metaphysics, algebras, maths, and logics apply extremely well to something beyond metaphysics: physics.
What is physics? It's those descriptions that are "descriptions that are not-only-definitions", which contrasts with metaphysics which is described only with definitions. The part of the description that is not-only-by-definition is everything else, which we call physics. Unlike metaphysics, physics does not provide us with a definiton for everything. We have no way of knowing if physical things by definition exclude other physical things. We may find that some physical things are reproducible. We may find, perhaps surprisingly, that formal metaphysics describes these reproducible physical things extremely well!
As the principle of contradiction removes indiscrimination from metaphysics, reproducibility removes indiscrimination from physics. Some formal metaphysics describe reproducible physics shockingly well. If our analogous formal metaphysical description of a physical description corresponds closely to the physical description, that's as close to a metaphysical immutable definition as we can get. All other options are farther away: not having a description at all, having an informal description which might violate the principle of contradiction, or having a physical thing that is simply not reproducible. The formal reproducible physics is the best description of physics we have.
Are there really no other methods get better knowledge of physical things? There must be other methods! What if we could somehow reveal an underlying metaphysical definition of the physical world? Then either that definitin is reproducible or unreproducible. If it's reproducible, it should be supported by our formal metaphysics of reproducible physics. If it's unreproducible, it's only knowable through singular revelation, which is informal metaphysics, which allows contradiction.
We have found that physics and our metaphysics may describe, through the principle of contradiction and reproducibility, everything that is describable which is not indiscriminate. We have found that without the principle of contradiction metaphysics is indiscriminate, and that within indiscriminate metaphysics "anything goes".
We have found that our best physical knowledge of the world comes from formal metaphysical definitions and reproducible physics.
Everything else is fundamentally indiscriminate to contradiction.
And that is how we know anything.