This article doesn't do a great job of explaining what probabilistic programming actually is. It's about 1) making machine learning and probabilistic modelling accessible to a larger audience, and 2) enabling automated reasoning over probabilistic models for which analytic solutions are inconvenient.
(Sorry for the wall of text)

The idea, in a nutshell: create a programming language where random functions are elementary primitives. The point of a program in such a language isn't to execute the code (although we can!), but to define a probability distribution over execution traces of the program. So you use a probabilistic program to model some probabilistic generative process. The runtime or compiler of the language knows something about the statistics behind the random variables in the program (keeping track of likelihoods behind the scenes).

This becomes interesting when we want to reason about the conditional distribution over execution traces after fixing some assignment of values to variables. The runtime of a probabilistic language would let us sample from the conditional distribution -- "what is a likely value of Y, given that X=4?". (in Church this is accomplished with query). A lot of models have really simple analytic solutions, but the inference engine in a probabilistic programming language would work for any probabilistic program. The semantics of this are defined by rejection sampling: run the program a bunch of times until you get an execution trace where your condition holds. This is really, really, grossly inefficient -- the actual implementation of inference in the language is much more clever.

An analogy to standard programming: it used to be the case that all programmers wrote assembly by hand. Then optimizing compilers came along and now almost nobody writes assembly. The average programmer spends their time thinking about higher order problems, and lets the compiler take care of generating machine that can actually execute their ideas.

Probabilistic programming languages aim to be the compiler for probabilistic inference. Let the runtime take care of inference, and you can spend more time thinking about the model. The task of coming up with efficient inference algorithms gets outsourced to the compiler guys, and you just have to worry about coming up with a model to fit your data.

Because you don't have to think too hard about the math behind inference, probabilistic modelling suddenly becomes accessible to a much larger subset of the population. A ton of interesting software these days is relies on machine learning theory that goes way over the heads of most programmers suddenly becomes accessible.

On the other hand, the people that do this work already are freed up to choose more expressive models and be more productive. The current paradigm is: come up with a probabilistic model, then do a bunch of math to figure out how to do efficient inference over the model given some data. Proceed to code it up in a few thousand lines of C++, and panic if the underlying model changes. The probabilistic programming approach: come up with a model, and write it in a few hundred lines of probabilistic code. Let the language runtime take care of inference. If the model changes, don't worry, because inference is automatic and doesn't depend on the specific model.

If you're interested in this, the Probabilistic Computing Group at MIT (probcomp.csail.mit.edu) has some interesting examples on their website.

An really simple example of Venture, their new probabilistic language: http://probcomp.csail.mit.edu/venture/release-0.1.1/console-...