The long oval in the center represents your submarine. The rest of the chart displays how far away and in what direction a contact is. Use tokens for contacts, as they will move.
As the sub moves toward the contact - and/or the contact moves towards the sub -the target moves into a new, closer range band. When the sub and contact move away from each other, the contact moves into a further off range band. As the contact comes closer, it should begin to resolve into separate contacts - first for the larger ships, then for the smaller ships. If your sub is in the middle of a convoy, having penetrated the escort, you might have contacts all around your sub.
The bearing is the relative angle from your sub to the contact. The chart is split into 90 degree quadrants, so zero degrees on the bow means straight ahead, while zero degrees astern means directly behind you. From zero, the degrees increase to 90 degrees to the right and left, then decrease back to zero. Right and left are always relative to zero on the bow, even when talking about bearings astern. So, to place a marker at "40 degrees on the right bow", find zero degrees on the bow and move the marker clockwise to forty degrees. For "40 degrees right astern", find zero degrees astern, and move the marker counter-clockwise forty degrees.
If the sub changes its own course, move the contacts the same number of degrees. For example, you get a contact at 20 degrees on the left bow. The captain decides to turn towards the contact to lessen the range and investigate. Move the contact to zero degrees on the bow and begin stepping the range in.