I really like these posts by Sanjeev Arora :
1) http://www.offconvex.org/2015/12/12/word-embeddings-1/
2) http://www.offconvex.org/2016/02/14/word-embeddings-2/
For a more theoretical explanation :
1) http://www.offconvex.org/2015/12/12/word-embeddings-1/
2) http://www.offconvex.org/2016/02/14/word-embeddings-2/
For a more theoretical explanation :
Though, some statements about analogies should be taken with a grain of salt, see: Tal Linzen, Issues in evaluating semantic spaces using word analogies, https://arxiv.org/abs/1606.07736.