When do you need to use broken shell extractor? When you have a case-head seperation, where part of the brass casing is still stuck in chamber. Since the base is now gone, you cannot use conventional means to get the brass out (i.e. close bolt and pull again, or use a cleaning rod to punch out the stuck casing).
How to use it? Insert the broken shell extractor like you are inserting a round into your chamber (but now it have the broken shell in it). Let your bolt slam home, then pull the bolt again. The broken shell extractor will act as hook, grab the inside of the broken casing and pull out the whole thing.
Avoid shooting unknown reloads usually will keep you away from the need to use this device.
One last thing: this question should be posted to trouble shooting forum, not ammunition forum.