Honestly, the idea of poor and clean going together is actually based on the idea that there is such a thing as a noble poor. For instance, a poor person who takes pride in their appearance, but can't afford anything better will have one set of 'nice' clothes that they keep clean for special events like, weddings, funerals, job interviews.
If you take the time to go to a country with a magnitude or two more people, say India or China, you'll find many many poor people that literally live in shacks with dirt floor. In the United States we have a magnitude of wealth above the rest of the world, as a "First World" Country.
When you have to worry about having enough food for you and your family each day, then all you can do is teach your children to be good people, and hope that they do not end up thieves, just trying to make life easier.
So... cleanliness is next to godliness is really nothing more than a mantra to remind yourself that being poor doesn't mean that you aren't a good person. It really is. My family was never very religious, but that statement is something my poor grandma lived by, and her house was always clean... no matter how poorly stocked it may have been.