I *think* this "pipe" is actually the core of a carpet roll (similar to the core of a roll of paper towels). But, I don't actually know for sure (it was being thrown away at a remodeling site); a thin-walled 4 inch PVC pipe is an alternative.
I currently have plastic shopping bags stuffed in the sides with a bolt to keep them from popping out when the pipe fills with water (I need to improve this but it does work).
1. Cut four 6" length pieces of a 3/4" PVC pipe. These four stubs will direct the water from the pipe into 4 of the top barrels.
2. Drill holes in 4 screwcaps to accept the PVC stubs. Don't worry about an exact fit (caulk will help later). Screw 1 screwcap into each of 4 of the top barrels
3. Drill 4 holes in the pipe that line up with the holes in screwcaps in the top row of barrels. In these holes, I fitted the 6" long pvc stubs and caulked them into place so they would direct the water into the barrels. Don't push the stubs too far into the pipe (you want the water to enter the stubs easily).
4. Next, caulk every crack so that no water can escape.
5. Cap the ends of the pipe (I used a big wad of plastic shopping bags with a 5" bolt pressed in to keep the rain from pushing the bags out). Leave a comment if you can think of a better alternative (the bags crumble after a season in the sun).
6. Cut a hole in the top of the pipe for your gutter spout to enter the pipe. Use a flexible accordian spout and put the spout into the hole. Stuff some netting (a bath scrungy works) into the hole to keep mosquitoes out.