Homie for ESP8266 is an ESP8266 for Arduino implementation of Homie, a thin and simple MQTT convention for the IoT. More than that, it's also a full-featured framework to get started with your IoT project very quickly. Simply put, you don't have to manage yourself the connection/reconnection to the Wi-Fi/MQTT. You don't even have to hard-code credentials in your sketch: this can be done using a simple JSON API. Everything is handled internally, by Homie for ESP8266.
You guessed it, the purpose of Homie for ESP8266 is to simplify the development of connected objects.