Swinging on plane in the golf swing refers to the golf club's angle of attack or the path it travels on the back and through swing. When teaching proper swing plane the club club should be swinging slightly under on the downswing relative to where it traveled on the back swing with little variation.
To swing the golf club on a proper plane first take a stance in with your back to a mirror or your shadow and and teach yourself by turning at a slight angle to the right if your right handed and left if left handed until you can see yourself.
Start swinging the club back until you see your club head swing behind past your back arm.
Stop at that point and check your swing plane by noticing where the shaft of the golf club is positioned. It should be angled up almost directly across your back shoulder or slightly over it.
If your club shaft is way over your shoulder your swing plane is to steep or vertical. Likewise, if it is down below or approaching your elbow its too flat or horizontal.
Repeat this step until you teach yourself to naturally swing the shaft to or slightly under over your back shoulder and hold this feeling and repeat it several times.
Then complete your back swing all the way to the top and hold. Now start your downswing and stop the club again at the same point to check your swing plane on the downswing. If you are on a proper plane the golf shaft should return to or slightly under where it was in relation to your shoulder on the back swing.