I'll answer the questions in reverse order, starting with the phonetic process and then talking about this specific example.
The name for this phenomenon in general would be elision; in fact, it also occurs in French. Elision refers to a sound being dropped or muted in the pronunciation of a word, and this often occurs to vowels in unstressed syllables. As an example, in the French word appeler, there is sometimes elision of the second vowel. It's difficult sometimes to give exact reasons for phonetic processes, but the reason for elision might be that these vowels are less important for understanding, so they can be left out when people talk quickly.
In the English word "prefatory", the stress falls on the first syllable, so the letter o is not stressed. In British English, this has led to the o being elided, so it is now a silent letter. However, the usual American pronunciation lacks elision. This difference applies more generally to the pronunciation of the endings -ary, -ery, and -ory in British and American English. However, it's hard to see a more general phonetic rule behind this; I would say this is simply one of the differences between British and American English that has developed over the centuries.
There are also words that usually have elision in American but not British English, such as the word "laboratory".