What's the meaning of 聊起 in 马小姐正在和她的好朋友李太太聊起这个男人的事情
It means &quot;to start talking/chatting&quot;.
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.
For example:

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 &quot;rise up&quot;. In your example and in this answer, &quot;to begin&quot; is figurative. The AllSetLearning link explains these two usages as well.