Compost, compost, how do I love thee?
SO DANG MUCH!!!
Ok, got that out of my system.
Brown/green: What they're talking about is stuff that has a lot of carbon (brown), like fallen leaves, dried grass, shavings from the coop, wood chips.
The green is stuff that is high in nitrogen, like fresh (green) grass clippings. Chicken manure, even though brown in color, is VERY high in nitrogen, as are most manures.
If you have all carbon, it will take a very long time to break down. If you have all nitrogen, it will stink. That's why the mix is important. The oft-mentioned ratios are 25 or 30 to 1, carbon to nitrogen. In other words, it doesn't take too much manure to help dried leaves break down.
If you build a pile of vegetative stuff, it will eventually break down. Think of the fallen trees and leaves on a forest floor. But for a garden, you need the compost to stay in its pile, and so that's why they sell compost bins.
You don't need to buy a bin. You can make a pile, put some welded wire fence around it, and that can be a compost bin.
You will hear the phrases "hot composting" and "cold composting." The stuff on the forest floor? That's cold composting. It sits there, various "things" from the dirt get in there (bugs, worms, fungi, a word that starts with "M" that I can't think of or spell*), and they break down the stuff in the pile, or on the forest floor. It been going on since there were forest floors. It takes a long time for fallen leaves to turn into dirt. Years.
Hot composting is the same only different, as we used to say where I used to work. If the mix of nitrogen and carbon are right, the pile is damp, and some air gets mixed in, the pile will heat up, literally. To 150F, sometimes. The stuff in the pile breaks down faster, and you can have usable compost in 3 weeks.
https://deepgreenpermaculture.com/2010/05/08/hot-compost-composting-in-18-days/
This site will tell you all about making compost, fast. Caveat: It's more manual labor. I do this in the spring with the kitchen scraps I've been dumping in my compost bin all winter, along with the poop I collect out of the chicken coop.
*Ah! Here's the word: mycorrhiza. Fungi in the ground that do amazing things with and for plants.