This requires a little bit of context, so please bear with me.
Ron Weasley: "Wingardium Leviosa!"
Hermione: "You're saying it wrong. It's Wing-gar-dium Levi-o-sa, make the 'gar' nice and long."
— Hermione Granger being condescending toward Ronald Weasley
In linguistics, one of the first things you're taught is that with a few exceptions (e.g. onomatopoeia, sound symbolism), the mapping between words and their meanings is completely arbitrary. To quote Shakespeare, "What's in a name? that which we call a rose by any other name would smell as sweet;". Nothing about the rose changes no matter what sequence of sounds you use to refer to it, and inversely, nothing about the sound sequence "r", "o" and "z" will tell you anything about the flower. Furthermore, contrary to popular belief, the specific signs of sign languages used by the deaf are just as arbitrary as spoken and written languages in their mapping between sound and meaning. In Harry Potter, however, the form taken on by spells does in fact seem to depend on the specific sounds of the incantation (as well as wand/hand movements). While spells do tend to have an etymology somewhat related to their function in other languages, Latin and Greek are no less arbitrary than English or any other language. What about wizards in other parts of the world who speak languages not rooted in a classical occidental language? Do Chinese wizards use the same incantations as English wizards? Superficially, it would make sense if the incantations and wand movements used are just traditional, the important part being a mental association between these actions and the effect that results. If this were the case, a young wizard could be taught to cast a levitation charm with the incantation Avada Kedavra, or Lumos, or Banana, or anything else.
"Lapses in concentration while charming can result in painful side effects — remember Wizard Baruffio, who said 's' instead of 'f' and found himself lying on the floor with a buffalo on his chest."
—The Standard Book of Spells, Grade 1
So what is it, the pronunciation or the concentration? Do the sounds spoken somehow influence or shape the magical "field" that is emitted? What about non-verbal spells? Has the wizard gotten skilled enough to mentally emulate whatever effect the sounds have?
One guess I have is the use of incantations as a means of "communicating" with a wand, which is shown in the last book to be something comparable to conscious and intelligent. Even then though, if the incantation were being used as sort of a code-word or instruction to the wand, whether wands come "preprogrammed" or learn the incantation concurrently with the wizard, this should be just as arbitrary unless specific sounds or thoughts of sounds are what create different types of magical effects.
Examples abound, the most relevant in this case is Sectumsempra. How is it possible that Harry could cast this spell correctly without knowing what it does, just from knowing the sound and wand movements involved? I vaguely recall something about wand movements next to it. The shape of a knife is not arbitrary, and used against someone will cause very specific, obvious effects. It's possible that the wand movements somehow directed the magic into a cutting "shape", but the incantation was still necessary, and it seems unlikely that if it were not arbitrary, Harry would have had to have had at least some idea as to the spell's effect.
The time Harry learned "Episkey" from Tonks brings to mind another possibility. I can't remember the exact quote, but he mentions that it's similar enough to other spells he already knows, and was later able to use it on one of the Gryffindor chasers. It seems that witnessing the spell was enough to learn it, but only because Harry had sufficient background knowledge. Possibly relevant,
"J. K. Rowling writes in Harry Potter and the Half-Blood Prince that Harry's knowledge tells him this spell could belong to a variety of healing spells, in the same way a species of plants belongs to a larger genus."
One thing from the books that seems very clear is that we know precious little about how magic actually works. It could very well be that there is a great deal more going on underneath the surface than what we can glean about the science of magic, but working with what is known, I've come up short trying to find an internally consistent hypothesis concerning the form-function relationship of incantations and spells.