Never used bridging reins..but I used to bridge my reins when jumping my horse because she used to snatch the reins so could just drop my hand and her neck helped restrain her...also help's you keep hold if you are prone to dropping a rein and keep you reins at the correct length...
A friend also bridges her reins on her bucking broncho horse because if she goes to far forward again the bridge goes to his neck which gives her support to get back in the saddle!!!
I regularly teach rider's who use the Mailer bridging reins and they are very useful for teaching young riders to keep a hold on the reins or are useful for those who let the reins constantly slip through their fingers, as the notches on them stop the reins from slipping easily. I find the brodge to be especially useful when teaching kids as they are prone to just dropping the reins and the bridge stops the reins from dropping completely.
I use them for jump schooling/training and at the sj shows.
I don't jump without them at home or at the show.
>And what's the purpose of them?
To keep the reins at the right length, and help keeping hands in better position. You can find out more about those reins on http://www.mailerbridgingrein.co.uk/
They help me out from the "oh, gosh!!!" situation, like, when jumping over one fence was not quite right as I wanted and I lost balance, I can still going without losing reins, and I can recover very quickly for the next fence which tends to come to you much quicker in those situations.
I don't get told "shorten your reins" from my jumping trainer these days!