Systems theory is an interdisciplinary theory that can be applied in many fields, such as science, mathematics, and business. It is the theory or philosophy that governs the operation of a particular system. In terms of science, it might be the laws that make up the function of human cells - the cellular system. In math, it could be the rules that govern the solving of an equation - algebraic system. In business, it may be the system of rules that govern how to manage an office.
The idea behind systems theory is that it seeks to explain how things work. Look at management as a career - if you are going to explain to someone how to be an effective manager, you first need to examine the system of management - what does it require? how is is successful? where can it fail? etc. Only by first doing this can you then explain it to someone else.
Here is a quote from the pioneer on Systems Theory, Ludwig von Bertalanffy.
There exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular kind, the nature of their component elements, and the relation or 'forces' between them. It seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles applying to systems in general.
In this way we postulate a new discipline called General System Theory. Its subject matter is the formulation and derivation of those principles which are valid for 'systems' in general.