I'll take a stab at trying to explain it. :)

First off, let's consider an economy with a single good (widgets), and no services. Any attempt to measure the economy as a whole (eg, to calculate GDP) is simply going to end up being the number of widgets produced per year.
Further, the total compensation for all factors of production is going to be equal to that same number, which in turn means that any employee's compensation (whatever unit of currency is used) translates into a share of the widgets being produced.

If everyone gets more efficient at producing widgets, the economy is wealthier (and GDP goes up). And each employee is (all else equal) receiving more widgets as their share. Conversely, if we assume that nominal wages have stayed constant, the price of widgets has gone down.

So that's a basic framework we have: Productivity gains = a fall in real prices = a growing economy = an increase in employee compensation. These are all equivalent things; different sides of the same coin. If we're getting better at making widgets, then widgets will be cheaper, which means we can afford more widgets for the same amount of (real) money. That's what economic growth is.

Second, let's consider an economy with two goods: Widgets and sprockets (and again, no services). Let's say half the work force is busy making widgets, and the other half is busy making sprockets. If we get better at making both, but we become much better at making widgets, then the price of widgets will start to collapse. Where 1 widget used to barter for 1 sprocket, soon it'll take 10 widgets to barter for 1 sprocket. Which is another way of saying that the price of sprockets will rise steeply.

That's Baumol's Cost Disease. If you have multiple items, and they experience different productivity growth, the ones which experiences relatively lower productivity growth will become relatively more expensive.

In addition, if you want to take the widget and sprocket example a step further, you might wonder what happens if productivity gains mean we're making more widgets than we strictly need. We'd probably see layoffs, as productivity gains mean fewer and fewer people are required to produce the widgets the economy needs. And of course the people left would probably earn very good wages, since they get to share the returns from the entire widget industry. Although given the high price of sprockets, they might not feel that rich. But where would the laid off people go? Well, into the sprocket industry; it's lower productivity means it can soak up more manpower. But with the returns from the sprocket industry split across more people, wages would be lower. Luckily, those wages would go further when purchasing cheap widgets, but high-price sprockets would be harder to obtain.

Which is, if you squint a bit, basically what we see, with widgets taking the place of the manufacturing sector, and sprockets the services secotr. US manufacturing output is high and (the financial crisis aside) keeps getting higher; US manufacturing employment is low and keeps getting lower. Manufactured goods are cheap and getting cheaper; services are expensive and getting more expensive. Skilled engineers command excellent wages, but struggle to find material goods to spend it on. Workers in the service sector suffer from low wages, although their material standard of living is still good. Etc., etc. Sound familiar?

Productivity gains are always a net good in total, however distributed. But the skewing effect of unequal productivity gains can be extremely wrenching. Telling someone that, actually, the machine shop they worked in is closing, but they can retrain as a hairdresser doesn't make people happy. Nor does pointing out that, even on the wages a hairdresser makes, they can afford a larger flat screen TV than the one they bought three years ago. They don't want a bigger flat screen TV; the last one is actually fine. They don't want to be a hairdresser with a giant flat screen TV; they want to be a machinist. With affordable health care. But that would require figuring out how to make sprockets as efficiently as we've figured out how to make widgets, and we haven't done that yet. (Although the US does seem uniquely bad at it, so maybe we can at least try to be less bad...)