In my humble, non-economist mind there are two forms of economics. The first is the academic/ivory tower form practiced by "real" economists from their offices, amongst themselves publishing papers and "studies" no one other than another economist would understand or even read.
It is of far more practical relevance and wide use than you suspect. The staff in central banks use research in empirical and theoretical macroeconomics to make and adjust forecasts and evaluate current macroeconomic conditions. All discussions in central banks draw from this literature, although it usually does so with some delay and it also usually add certain features to DSGE models before using them to suit their needs. It is also part of the standard toolkit of the staff in finance departments that deal with budgeting issues. This is also useful for private businesses. The most obvious example comes from finance: a firm involved in fundamental trading strategies using macroeconomic data definitely pays attention to the literature on asset pricing, volatility, forecasting methods, etc. Hell! How many firms use the Fama-French factor models? Another less well-known example is the set of rules business managers use in the real world to evaluate the profitability of an investment project. All those rules (such as the net present value and more sophisticated real option pricing) are derived from microeconomic theory, even though the quickest methods that are most often used with small heuristic corrections date back to the 1950s.
Finally, it's not impossible to understand at all. The most basic DSGE model you can imagine is the Ramsey-Cass-Koopman growth model with technological shocks, i.e., the first real business cycle (RBC) model of Kydland and Prescott. The basic stories behind the equations are easy enough to understand:
1. Business cycles are caused by fluctuations in the production capacities
2. As opportunities for working and for renting capital vary, so does income, but people prefer stable consumption paths (across time and states of nature);
3. Because (2), some of the volatility of income (GDP) is shipped into savings (and therefore investment) to allow for smoother consumption paths.
Using microeconomic studies, you get very stunning performance: you can match the volatility of output growth, investment growth and consumption growth, as well as order them properly. The same goes for hours worked. The maths is, however, necessary if you want to know what's wrong: it's not obvious from the above stories, but to match those empirical facts for the 1950-2000 period in the US, you need parameter values and functional forms that imply ludicrous things. For one thing, you have to be able to accept a very elastic supply of labor, even though microeconomic estimates show it's very small for individual people. You also have to be ready to live with a process for technological growth that is so persistent and volatile, it implies a 37% chance of technological regress every quarter.
Of course, that was 1982, if I recall. Today, models are more sophisticated, but the fact that a silly looking little theory could do so well and the fact that we could check so many details about our theories if we bother translating our intuitions into mathematically meaningful terms led DSGE models to become very widely used. There is nothing arcane or remote about this. It's the same damn thing everybody does in their heads: pick a handful of salient factors influencing behavior and run some kind of mental simulation to answer questions about "what if" kinds of scenarios.
That realm spends their time writing and arguing arcane theories and hypotheses and embracing the dogma of various "schools" and debating the merits of each over snifters of brandy at the faculty club. The cliched "if you lined up all the economists in the world end to end they wouldn't reach a conclusion" pertains.
That is an impression you can only get if you never saw economists talk about their research. Suppose we pick a theme: the minimum wage. We have statistical techniques that can potentially identify the effect of a specific minimum wage hike (we have many of them, some using microeconomic data and some using macroeconomic data). We also have many theoretical models we can calibrate or estimate to see if the results we get from statistics seems to emerge in simulations when we take into different factors that may play a role. Sometimes, after all of this is done, the conclusions converge. When you try to make sense all of these different pieces of evidence, you at the very least end up rejecting certain ideas if not end up picking just one answer. That's what we actually do: argue over evidence, argue about how to modify models to see if intuitions make sense when you cross all T's and dot all I's.
