In a nutshell starting from the rod end you need a slip-bobber float string, a bobber float stop bead, a float of some kind, and a series of split-shot sinker weights to keep your line, hook, and bait presented as vertically as possible.
The bobber string stop can be slipped up and down the monofilament line to allow you to fish very precise depths.
When you see a current stream estimate the depth and set your string stop so it lets the bait float 6" or so off the bottom. Cast into the stream and keep your bail open, feathering out line and watching for the bobber to disappear. Retrieve once the bobber gets too far away to see well and send it down again except maybe raise the string stop again a few inches to work the same path a tiny bit deeper.
If you set the depth too deep or the bottom depth changes your bobber will slowly sink in the current. This is due to the weighted line hanging up on the bottom. Grab the line with the fingers you are feathering it with and give a pull. This will pull your line away from the obstruction and hopefully settle down in the current on the other side.
You wade downstream using this technique since you are relying on the current to float your bait downstream.
Oh, I almost forgot the all important answer to the question "So how do I catch a fish this way?".
When your bobber goes down or when it goes sideways unexpectedly you close your bail, reel in some slack, and set the hook.