The interesting thing is that whilst people usually do connect with those they know well (at least most of them), they often fail to leverage those contacts effectively.
To be honest, connecting with people that you already know has little value unless you use the power of LinkedIn's visibility to discover who they know.
Once you have connected with someone, it is then possible (assuming their settings are at the default level) to see who they are connected to.
This can be highly revealing, and may show you that a connection could be in a position to introduce you to someone you'd like to do business with.
Simply click on a connection's number of connections (see pic below) to reveal who they are connected to.
Once you have found someone of interest, just pick up the phone and ask your connection if they would be happy to introduce you. Asking the question “Can you introduce me to X?” is a much better question than “Who else do you know?”
This is where the power of LinkedIn really comes to the fore.
Alternatively, LinkedIn also provide a facility that allows you to send a message to the person you wish to be introduced to via your mutual connection. This is called a LinkedIn Introduction.
LinkedIn introductions can be a highly effective way of contacting prospects for the first time, but there is a knack to making them work for you. The video below explains all: