By identifying a sentence’s grammatical constituents and the relationships interconnecting those constituents, a diagram shows how the human mind analyses a sentence’s underlying syntax. It illustrates the grammar that holds the pieces of sentence together.
The particular notation for showing this structure is much less important than the basic act of doing so. Early students usually start by distinctively indicating the sentence subject and the sentence predicate, perhaps with something as easy as a simple underline for the subject and double-underline for predicate. Even with as simple a diagram as this one can now teach students how to exchange one subject or predicate for another, or how to invert the two.
From there you move on to other structures like noun phrases, verb phrases, subordination and coordination, dependent and independent clauses, prepositional phrases, and all the rest of them. Notice that we are not talking about parts of speech here but rather of higher level syntactic units.
This is what matters. What particular graphic representation you choose is much less important. You’re trying to teach the students that syntactic constituents exist, how to recognize them, and what they can do with the them.
The English word syntax derives from the Latin syntaxis, a transliteration of the Ancient Greek word σύνταξις, which per Wiktionary is composed of the components syn- meaning “with” and táxis meaning “arrangement”. Syntax is one of the two pillars of grammar (the other being morphology), because it shows how language fits together.
Without arrangement, words alone can only convey extremely simple ideas; there is no language. With arrangement — that is, by taking syntax into account — an infinitely richer set of possibilities exists. And this now is actual language.
Our close cousins the chimpanzees can learn words, but it appears that they cannot learn that fitting them together in a particular way matters. In contrast, dolphins are able to process language at this second, higher level, and therefore can understand the difference between “take the ball to the ring” and “take the ring to the ball”. That’s what happens when you add syntax to words: you get language.
Syntax matters. It’s part of what makes us human. Without syntax, there would be no language as we know it. Teaching how to diagram the underlying syntactic constituents of a sentence brings an understanding of what language actually is.