It's pretty preposterous that a sound alone could summon forth magical energies, especially when you start to think about what would happen if you recorded an incantation and then played it back. Or, even in a more low-tech example, if a parrot learns a few spells, can they turn the bars of their cage to rubber and their owners into statues of sunflower seeds?
The answer to this, unfortunately, is no. This wouldn't really work in very many stories. What this means, then, is that there's something about humans that makes magic work. Where this gets interesting, though, is that one of the many things that makes humans superior to beasts of the air or radios is the well-developed speech centers of our brain (like Broca's area). There's a lot going on in our heads solely devoted to figuring out what people are saying, and then figuring out what to say in response (to realize how hard this is to do, just try holding a conversation with Siri, or a dog). As a result of many generations of evolution, we're now the best species in the world at talking, and expressing complex ideas, using the world's most highly developed brains.
Thus, it's pretty simple to say that whatever lets us do magic sits right amid those speech centers of our brains. There are some really interesting brain injuries where people become unable to speak(look into Broca's aphasias); a sort of reversal of this could explain why most people can't use magic. As for how speaking the words actually leads to the effects, think of it like learning to move; syllables and phonemes are like nerves, flexing and extending magical muscles. With practice, eventually you can learn what words lead to the right magical movements to produce useful results; once you know the words, though, you can teach them to others. Plus, continuous use of your magical speech skills should lead to a better understanding of how things fit together, and you should be able to formulate more complex magical sentences to produce even larger and more nuanced spells.
The point here is that it's not the words that have the power, it's your brain (or, by extension, your soul, if your world has souls). Words are more like an involuntary reflex, like how some fingers can't be moved on their own without moving the other fingers with them, or how British people insert 'r' sounds between some words. With practice, spellcasters should be able to fire the right neurons in their speech/magic centers without actually pronouncing the words, but again that would come with practice.
As for what this language would sound like, its structure/syntax/etc., that really depends on how the magic works. This is really just more of an interface; just like how our brain interacts with our bodies through the cerebellum, it can interact with magic through the Broca's area.