Why is the "A" in some adverbs silenced?

I'm an English learner, and I have had this question troubling me for a really long time.

In the word "magically", we do not pronounce the "a", so it is pronounced just "magicly," like most adverbs having the "-ally."

But in the word internationally, we pronounce it as its full form. So is there a rule for this and are there any words which are similar to "internationally"?

In a comment, John Lawler wrote:

It's not just the "A"; it's any unstressed vowel, however it's spelled. Unstressed vowels can be spelled with any vowel letter, since letters don't determine sounds, or stress. And it's not just adverbs; it can happen to any word. There are a lot of rules, but one stands out -- in a long word the unstressed vowels will be reduced, centralized, and/or deleted. Listen to the way native speakers pronounce them in natural speech.

It's rarely mandatory to pronounce -ally as one syllable

In the word "magically", we do not pronounce the "a"

Not necessarily. As Michael Harvey said in a comment, some speakers do have a four-syllable pronunciation of magically. The Merriam-Webster entry for magical indicates that the pronunciation of magically can be either /ˈmædʒɪkəli/ or /ˈmædʒɪkli/.

so it is pronounced just "magicly" like most adverbs having the "ally"

This generalization is a bit too broad. Magically falls into the more specific class of -ically adverbs, which commonly have pronunciations in /ɪkli/. But as you've found, there are other -ally adverbs (such as internationally) that are commonly pronounced with /əli/. In fact, I wouldn't say internationally is an outlier at all.

I think a reasonable rule to follow would be to use /ɪkli/ in -ically adverbs, /fli/ in certain -fully adverbs (such as beautifully or awfully), and /əli/ in any other adverb spelled with a vowel letter followed by -lly.

Many -ly adverbs have optional shortened pronunciations

As John Lawler said in his comment, many long words have shortened pronunciations that drop an unstressed vowel. I wrote an answer to a previous question about the pronunciation of the adverbs "personally" and "finally" that has some information about this.

As the comments beneath your question indicate, internationally may be shortened by dropping the vowel after /ʃ/. It may even be possible for some speakers to drop the vowel between the /n/ and /l/, even if that pronunciation is not listed in dictionaries.

But these kinds of optional shortenings are often variable and difficult to perceive or precisely describe, and so for a non-native speaker, it may not be very effective to try to reproduce these pronunciations. So I won't try to give a rule about where or when those kinds of reduced pronunciations are likely to occur.