1. What accounts for the productivity gaps? It's largely education and so-called "human capital."
2. So how is it that I can personally increase my wages by moving to a different place? It depends on who you are, but for most people it's typically because you'd be working for a different richer set of people. This is the basic reason why moving from Mexico to clean hotel rooms in the United States to be a maid can increase your income. It's not that you have better skills in the US, it's that there are more rich people who can hire maids.
3. But isn't this really just the higher cost of living? No! The increased productivity of workers in the Boston and San Francisco MSAs is partially cancelled out by high real estate prices, but this is less true in Chicago and even less true in Minneapolis. What's more, this is a real currency union. A dollar may not "go as far" in Seattle as in San Antonio, but their both dollars. Employers aren't running charities and don't pay high wages to people in high cost areas to be nice, the wages are driven (roughly and imperfectly) by real differences in the value of the outputs.
The moral for the US is a story I tell about housing policy in The Rent Is Too Damn High. The story about Europe is different and more about language and culture. Inability to speak Dutch or German or Finnish puts real constraints on what jobs you can obtain in the Netherlands or Germany or Finland and relatively few foreigners speak fluent German and approximately nobody speaks Dutch or Finnish. If all residents of the European Union could magically come overnight to speak the same language, labor mobility would go up, people would rearrange themselves, and overall productivity would be much higher.