Why is the study of grammar so important? It is because the rules of grammar, if not absolute and eternal, are nevertheless based upon the framework of reality--at least in so far as we understand reality. To be correct grammatically does not necessitate flawless understanding of reality, but real ignorance of and errors in grammar (not simply careless typos) will affect our understanding of reality. Grammar errors can mean errors in understanding.
...How does actual diagramming help? It trains the mind the recognize the parts of speech and other structures of language, making explicit and visual what is implicit and mental, and thus familiarizing the diagrammer with the structures of grammar, with the structures by which we understand reality. Knowing those structures well will help readers, especially in our hyper-visual age, to understand the meaning that is being expressed through language. It helps particularly with sentences written in less-than-familiar construction, such as "Now is the winter of our discontent made glorious summer by this sun of York," a sentence a proficient diagrammer will see immediately as "The winter of our discontent is now made glorious summer by this sun of York."
Apropos of nothing, I was flipping through a cookbook this evening and came across the entry discussing a style of ham from Spain: jamon serrano, or mountain ham. In Spanish the adjective follows the noun, and I wondered idly if jamon serrano and serrano jamon had such disparate meanings as the corresponding English phrases. There's quite a difference between "mountain ham" and "ham mountain".