It's not really about the ammonia (although that is an issue), it's just about them not having enough space to move about in. And a smaller body of water will get dirtier quicker. Also, most people don't even give their betta's a filter. And while betta's can handle dirtier water than a lot of...
They're called fighting fish over here. Just thought you should know, most of the tanks advertised as betta fish tanks are not suitable for any kind of fish. I saw that they sell them in cups in Walmart. That's disgusting.