You will want to see https://help.twitch.tv/customer/en/portal/articles/1302780-twitch-irc for information on connecting to TMI (Twitch Messaging Interface)
There are several projects available on sites like github that could provide chat logging abilities. A quick google for “Twitch log bot” should yield lots of possibilities.
PS: you can connect/read the chat (read only/anonymous mode) by connecting as “justinfan00000” using a random number.