Coaches say "stay connected" constantly and rarely define it. Here is a usable definition: connection means your arms and your torso move as one unit rather than independently.
What that looks like
In a connected swing, if you stopped the body turn, the arms would stop too. The chest turns and the arms and club go along with it. In a disconnected swing the arms carry on by themselves, which means the club can end up anywhere.
Why it matters more than almost anything else
Consistent radius
Connected arms keep a constant distance from your sternum to the clubhead. Constant radius gives a constant low point, and a constant low point is what separates crisp iron players from everyone else.
Sequence
Big muscles are slower and more repeatable than small ones. If the body drives the swing it repeats. If the hands drive it, the swing depends on timing, and timing is the first thing to fail under pressure.
Speed that arrives at the ball
Speed created by throwing the arms peaks too early. Speed created by the body turning while the arms trail behind arrives at impact. Same effort, very different result.
A thirty second test
Cross your arms over your chest and turn back and through. That is what connection feels like. Now hold a club and make the same turn. Did your arms do anything extra? Most golfers find they lift.
What connection is not
It is not clamping your arms to your ribs and swinging like a robot. The arms do move relative to the body, especially in the backswing and through the release. Connection means they move with the torso rather than instead of it.
How it gets trained
Not with a swing thought. With a constraint, repeated, then removed. That is the whole logic behind connection trainers.