Broach means to bring up or introduce a sensitive issue. If your best friend has severe phobia of spiders, you might want to delicately broach the topic of your new pet tarantula, Mr. Fuzzy.
Let's say you want to go on vacation with a friend and you ask your dad because he is more likely to say yes. He will probably tell you that he will broach the subject with your mom and let you know. In a less common (and older) usage of broach, if you put a hole in something in order to get out what's inside you broach it. The piercing tool you use is also called a broach. Think of piercing someone with your idea the next time you broach a touchy issue.