Without looking up the exact specs of the ESP8266 I can't tell you how long the battery will last but it does have a very powerful WiFi so it is likely to use a bit of power to run it. Anytime you transmit a radio signal your using lots of power basically your sending out power through the air, that's how a radio transmitter works.
If you want to go wireless then your always going to use more power.
Yes he talks crazy fast, U will need to watch it bit my bit and keep stopping the vid and replaying but if you google around you should find lots of tiutorials to do this sort of thing as internet of things is the biggest crazy of 2017
You would be able to google around and find other boards beside the ESP8266 that do the same thing, it is just more of a case that the ESP8266 was super popular because of price and amazing WiFi reception even at big distances from router (this great reception is likely to mean higher power consumption for the ESP8266 over other chips around)