It's actually very simple: it means that the right hand (which was originally playing in the top-most clef) is to play the half note pointed to in the lower clef.
So basically, you are supposed to play the chords in both your hands, then your left hand jumps down to play the octave on the
D, and the right hand goes down to play the
D below middle
The line that goes back up shows that the right hand moves back up into the higher clef to play the chord in the next measure.
Perhaps this is a comment and not an answer.
In music scored for the Barbershop style, such a diagonal line indicates that the melody is passing from one part to another. Reasoning by similarity, the diagonal line may indicate that the melodic line moves in an unexpected way, which could alter the interpretation.