As the chart suggests there are a certain number of seats/rooms sold at each bucket or fare. Each fare gets a letter code. Y is usually the most expensive. I believe the next fares goes something like YA, YB, YD, and YE being the cheapest. (or something like that. I use to know the system better) When a train goes on sale, usually all the buckets are open for sale. There might be 10 YE buckets available. Once those sell out, it bumps up to the next bucket, YD and so on. Sometimes Amtrak will manipulate what buckets are available though or place restrictions on some of them. On popular travel days, when Amtrak knows that people might pay more, they may open seats at a higher bucket. Same thing will happen on Saver tickets that have certain restrictions. That Saver fare might require 14 day advance purchase, so within 14 days that fare will be cutoff.
Things get complicated when trains obviously serve over a dozen stops though, but in actuality, Amtrak's fare system is pretty simple. Airlines use the same system on a huge scale.