Connecting is a two way street. When you hover over the name of a community member in a piece of content, view a profile, view the member directory, or view the members tab of a community, you will find the ability to connect.
If the member accepts your request, you will connect and they will be visible in you contacts on your profile. This is a good way to collect important contacts and have them at the ready for consultation, engagement, and the sharing of ideas. When setting up your profile security, you can also allow contacts to see more about you than the public or authenticated users.
Note: If you accept a connection request, that member can directly message you and you them.