The simplest way to regard them is just as a programming language. Such a language contains just one construct - the rule - and so should be easy to learn; unlike (say) Pascal, with its range of data types, loop structures, conditionals, and other statement types. There have in fact been several production-system languages, most used from the mid-70's to the mid-80's. The best-known is OPS5, promoted for writing expert systems.
Although it's easy to understand how production systems work, it is not in fact easy to write big programs, and production-system languages have fallen out of favour. The problem is that it can be hard to see exactly what the resolution strategies will do, and hence in what order rules will fire and data be added to memory. I have added some features to the Eden production system which avoid some of these problems.
The main reason for being interested in them is their use as cognitive models. The notion of a production system probably dates from the 40's, when some logicians used them as idealised mathematical models of computation, in the same spirit as a Turing machine. As far as psychology is concerned, they became important in the early 70's, when Newell, Simon and Shaw adopted them as a model of human cognition. The idea was that, at the lowest level, the brain is of course built from neural hardware. However, this hardware implements short-term and long-term memories, rule matchers, and other components of a production system, in the same way that the logic gates in a computer implement adders, registers, and other components of a digital computer.
We can understand how a computer program works by explaining its operation in terms of adders, registers, and so on. It is not necessary to know how these are built from logic gates. In the same way (say the production-system theorists) we can explain cognitive processes in terms of the rules that are fired, the symbols that are added to STM, and so on. It is not necessary to know what is going on at the neural level.