Why does averaging a sentence's worth of word vectors work?
It works for the same reason why the good old bag-of-words + TF-IDF works. Despite loosing some word ordering information, a text can be still classified by the typical keywords. Since texts on different topics differ a lot with respect to the vocabulary used, simply putting together the words' embeddings might work surprisingly well. 

Here is a paper that shows that a simple sentence embedding methods beats sophisticated supervised methods including RNNs and LSTMs. Their method is just a weighted average of the word vectors, modified a bit using PCA/SVD. Section 4.3 tells that the word ordering plays a role, but not too much.