One of the essential tensions of being a software developer is the fact that software isn't really useful on its own. It has to do something. That thing it's doing is the domain. Whatever very-specific information that the software system is organizing and passing, more specifically.
For example, I maintain an e-commerce system. More specifically, I maintain an e-commerce system built for a cosmetics company. E-commerce is a field, cosmetics is a field, cosmetics e-commerce is specifically the domain I need to become an expert in. The more I can differentiate cosmetics e-commerce from other kinds of e-commerce, the more well-tailored I can make my app. And the more well-tailored I can make this app, the better I'll be able to understand the more general case of making custom e-commerce systems for other industries.
The more I go through this process of understanding the specific needs of the industry, the more domain expertise I'm building. Domains can subdivide as much as industries subdivide. Say the cosmetic industry started to split along product lines, then I might start gaining insight into how lipstick companies operate, and be able to tailor a product offering straight to them.
The domain doesn't have to mean commercial industry. You might have domain expertise, say, in XML parsing. Then you could wax poetic on the differences between Nokogiri and say, Chrome's internal parser.
So, essentially, it just means "expertise". More specifically, the type of expertise that allows you to customize a software system around it. Which can be in practice quite different from normal expertise. The guy who devises our makeup formulas is definitely an expert.
But he can't design the system to actually manufacture it. The guy who writes those systems might not understand the actual formulas, so the two types of experts have to work together.
When startups talk about the need to be a domain expert, it means getting out of just pure programming and into the actual meat. Rather than hire an expert, to become one yourself. This not only allows you to reduce staff, it also makes it easier to make better, more custom solutions. The Soylent guys have to make the system to manufacture their product, but they also must understand nutrition, food, the regulatory environment of the food market.