While this doesn't speak to the etymological reasons for why 'e' specifically is the most frequent letter, there is actually a very interesting statistical reason that there is such a letter: Zipf's law.
Given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table.
Zipf's law was originally discovered with regards to Word frequency (e.g. the word 'The' is the most common word in the Brown Corpus), but also applies to many frequency and ranking comparisons, especially where humans are involved in the ranking process.
Essentially, the easiest to recognize, most generally useful sounds and concepts are used more often while the more specific, harder to articulate sounds and concepts are used less often. This generally follows a "long tail" logarithmic curve.
As for the letter 'e', it often represents very easy to pronounce vowels, and orthographically it is also used in English orthography to "silently" affect another vowel (e.g. 'ate' vs. 'at'). Added up, it's a pretty useful little letter!