How do_action funtion works with path parameter?

The first argument of do_action is the $tag, a string. As far as the API is concerned, there’s no special meaning to the contents of the $tag, it’s just a unique name that’s matched as a whole to hooked actions of the same name.

The structure and naming convention of tags is up to the plugin developer. The example you provide may in fact be a path, but whatever purpose that has is going to be specific to that plugin. The possible values of $part and $slug and the event that triggers the action is information you will need to seek from the plugin’s developer / support.