The magic is alive by itself
Magic isn't so magic as everybody thinks. Maybe magic is produced by an omniscient entity, such God or the weave, or is produced by a collection of minor gods, spirits or demons.
Those creatures hear the calling of wizards and perform the magical effects according to the set of instructions given by the spell incantation.
Those primordial creatures exist from the beginning of time, and only speaks a single (now dead) language. That is why spells use that language, they use the language of the person who made it / who produce its effect.
Maybe it's just a blessed language. During the time of creation, gods made this language and blessed it with magic, hiding power on each word. New, alive languages lack this blessing and so they can't be used to cast spells.
It isn't a language, but a collection of magical sounds and instructions
It's told to be a language, but that isn't true. These words, or for being exactly sounds and written drawings had inherent magical properties. They're used to channel, unleash and command hidden sources of power from the very core of reality reprogramming the very laws of physic. Through a deep, large and precise research, wizards from all times have been collecting and cataloguing those sounds and symbols (or commands), carefully tailoring and crafting spell's incantations (or scripts).
No one knows the meaning of those ancient words if they have one. The only current documentation we have are the spells itself, the rest was lost during the ages.
It's inherently magical
Similar to both ideas above. Speaking in your every day with a magical language is extremely dangerous. You wouldn't like to accidentally unleash the anger of a thousand of demons or to change the fabric of reality with a single "Hi".
It isn't an spoken language but a mathematical or programming language
Magic can't be done using such ambiguous languages as we use to speak. An incantation must always mean the same no matter the age or context. That is why the first wizards made an artificial language, a perfect template to make blueprints for new spells. Such annotation language is specifically designed to make spells, and so it's unspeakable for everyday uses.
For example, programming languages are -like the name suggest- languages, but they are used to make programs, not to speak. You can make a program in Python or C#, but not speak your every day on it.
Magic languages have the same effect, they are used to describe the behaviour of spells, not to talk. That is why no one can translate magic languages to normal speech and retain its power. You can turn a C# script into pseudocode or even plain English, but the compiler won't compile, as the spell won't be casted.
It requires an exact structure, pattern and meaning
El árbol volador es verde y esta junto al río. Esta lloviendo.
The flying tree is green and it's next to the river. It's raining.
Look the differences, "árbol volador" is translated as "flying tree" instead of "tree flying". "El árbol" becomes "the tree", which lose the grammar genre. Both "es" and "esta" are turned into "is", which is context dependent. "It's raining" use a neutral pronoun, which doesn't exist on Spanish.
All this slightly changes in meaning produce a hugely side effect on spells, which is too dangerous.
It helps mages to maintain focus
Reciting a completely memorized incantation helps to maintain concentration. Reciting something in a language you know allows you to understand it and become distracted. Incantations don't mean anything, they are just techniques to help spellcasters to focus and channel energies. If you accidentally think in anything else, the spell may go wrong. That also explains why each spell has so different phrases, to avoid commit mistakes.
It requires a language that can't evolve
Scientific use Latin to make scientific terms because it's a dead language. Since no society uses it on the present day, the language will never change. No collocations will be made not words will change their meaning.
If you use a language which words changes over time (i.e: Any non-dead language) effects of spells will also change over time, which is something that everybody wants to avoid. Maybe the hotness aura of yesterday produces an insanely hot aura which burned everything, while today it produces an aura of extreme beauty.