It was golf instructor Jimmy Ballard who made the concept of "connection" popular. The idea is to keep the target arm in contact with the chest so it moves with the chest and not independent of it.
Your target arm (left for right-handers) is the radius of your swing, and it controls the position of your clubface. If you keep your radius intact from setup to impact, your ball-striking will be very consistent. Unfortunately, most golfers change the radius as they swing and are forced to make some sort of adjustment to get it back to where it's supposed to be.
To feel the connection, try this drill: Place a head cover under your target arm, then keep it there with light pressure all the way to the top of your swing and well into your follow-through until your connection is naturally broken and the head cover falls to the ground.
Use this head cover drill when you practice, then when you play, plant your target arm on your upper chest at address and leave it there. Imagine that there is a strip of Velcro on your chest and a strip on the underside of your target arm, locking the two together. If you break the connection, you'd hear the famous Velcro rip. Your job is to swing without any ripping.