from the phrase I would expect that "aesthetic promise" is similar to a monetary promise, except as applied to aesthetics instead of money. A promise that something will be given.

from reading the article "aesthetic promise" seems to be "that particular bit of aesthetic satisfaction that you counted on when starting out on the series", in other words, one of the aesthetic promises of a continuing series of books is that there is a conclusion, so you read one book and the next, expecting that at some point they will all be put together into a whole.

Rather like how readers of Dickens day started reading his serialized novels in their papers expecting that the novel would in fact have an ending.