This device is a key part in things like my auto cobble factory (inspired by/copied from things I've seen in these forums) and systems with a piston pushing a block out of the way of a DB.
Here's a picture. Should save me 1000 words.
The way I used this in my cobble factory was to give a quick pulse to the BD that had just gotten dispensed from the one above it. If you place a BD next to an existing power source, it doesn't actually turn on, (bug?) so it can't turn off and suck anything in when the power source is removed. Because of this, I needed to have a pulse power it on and off quickly right after it was placed, since having the power on already when it was placed didn't work, as I said.
The need for a quick pulse should be obvious for the second situation I mentioned. If the DB is hooked directly to the piston, then after the piston pushes the block that was in front of the DB away, the piston doesn't retract, since the DB is now seeing the extended piston head. If you hook this system up to the device so that the DB is the input and the piston is the output, then when a block is placed in front of a DB, the piston extends and retracts quickly, and so it doesn't keep the DB triggered like before.
Edit: Playing around with it, I found that the repeater is not needed, as the the delay of the NOT gate is sufficient.