You can read them, grep them, search for links, nicknames, everything. Now I am used to using IRC (Internet Relay Chat), which is kind of oldschool but – depending on the chat client configuration – can log everything to text files. Sometimes you want to re-read some conversation you’ve had, or search for something the chat partner has said, like a link to a website. However, a major downside for me is the inability to export your chat. This is a thing I like about Hangouts, it happens on all my devices and I can hop in and out of a conversation wherever I am at that time.
On my smartphone, I mainly use WhatsApp and Threema, but both lack other clients so far WhatsApp web can only be used from one device at a time, and Threema lacks a web/desktop client, so communication is limited to using the smartphone which is somewhat annoying for longer conversations. Google Hangouts as messaging platform has some advantages over other often used services. Click this link and follow the instructions. TL DR: I’ve written a web tool you can use to view and download your Hangouts chat history.