The slopper strap has two purposes. ONe is that in older days, bridle reins might be made of rawhide braided laboriously. So, if you had a part near the bit where it suffers getting wet and other abuses, it can be replaced rather than the whole rein, this would save work.
And, when you use heavy rope reins, as many current Western riders do, the rein has more "feel" to it, due to it's weight. When the rider just lifts the rein, the horse will feel a change and know that a cue is coming. The slobber strap, by virtue of being wider, makes a wee bit more resistance to the metal ring, so when the rider picks up the rein, the horse feels a bit more vibration through the bit. And, the added weight right up at the bit itself, makes the rein fall immediately back into the "nuetral" position when the rider drops the rein, so the hrose gets a very noticeable and immediate release. Helps with timing and training.