I tend to think of the different forms of ancient Greek that we are aware of as being more like regional variations than wholescale new languages. Differences tend to be on the level of vowel/consonant changes and some differences in case endings, rather than wholescale alteration of vocabulary or syntax (which doesn't really change much at all between the dialects we know). There's an excellent, even surprisingly accessible summary of most of the morphological changes in The Ancient Languages of Europe by Roger Woodard.

The most obvious example from my view of the intelligibility of dialects - particularly for an Athenian - would be Greek tragedy, where the text shifts dialects from episodes (in Attic) to odes (in Doric), yet the audience would presumably have had no problem understanding the shift. Bonner's Mutual Intelligibility of Greek Dialects, while unsurprisingly Athens-centric, illuminates a whole raft of examples of communication in different dialects without difficulty.