It means "to start talking/chatting".
起来 can be used as a direction complement to express that the action has started.
As it's common in direction complements, when there is an object, the second direction verb 去 / 来 can be moved after it, and it can even be omitted1.
- 咱们动起来吧 = Let's move！
- 我刚看起这本书来，就觉得无聊 = I just started reading this book, and I'm already bored
Additional examples can be found here.
1: in the linked question, 起来 was used in its more literal meaning of "rise up". In your example and in this answer, "to begin" is figurative. The AllSetLearning link explains these two usages as well.