Tversky & Kahneman (1974) asserts that it is so, and some scrabble-playing experience convinces me that it's true. However, I've never seen an explanation of why it is true. I assume it has something to do with how the lexicon is organized, but I haven't been able to find any sources on that point.
This is a very interesting question. Unfortunately, I was not able to find something that would give you a clear answer.
In essence, I think this question is asking for a cognitive mechanism underlying word generation in phonemic/phonological verbal fluency test which is a matter that has rarely been addressed (Robinson et al, 2012). Studies such as the previous and this have revealed that the neural systems for word generation by letter and by semantic cue are different, so one can assume that the explanation for the phenomenon you mention is probably not relevant to semantic properties of the word.
Also, studies by Troyer et al (1997, mentioned here) have shown that in a phonemic fluency test, a person often uses a strategy called "clustering". In this case, it is phonemic clustering and "clusters are defined as successively generated words that begin with the same first two letters, differing only by a vowel sound, or are homonyms". This may be indicative of how words are organized and represented in verbal memory.
However, the question remains as to why this form of organization and representation occurs instead of another. A possible explanation may lie in the ways we use to learn words, namely in what has been named as statistical learning. This correlation I am making is certainly a speculation and I don't know if it is a valid one but I imagine that because syllables form into words according to the transitional probabilities from one to another, a phonological representation of a word might be more easily activated by the first letters while a parallel activation might also occur throughout a certain phonemic cluster for successively generated words.