When you chew the fat with your friends, you sit around chatting and gossiping with them.
Nothing too important gets discussed when you chew the fat — it's most often a light, social kind of talking. The expression may be related to "chew the cud," which means to talk aimlessly, ruminating over the same subject in a casual way, like a cow chewing its cud. It also seems that chew the fat evolved from an original nineteenth century British meaning, "to grumble, grouse, or complain."