The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar graphs and so one can use planar geometry (say, the Jordan lemma) to investigate the word problem in complicated groups. Similarly, asymptotic cones are useful in geometric group theory because they allow to study large scale properties of a discrete object (a group) by looking at small scale properties of a continuous object. I would like to know a similar answer for operads. 


 **Update** Many thanks to everybody for your answers. Unfortunately I can accept only one. So I just accept the first answer.



Why are operads useful?
Here are a couple 2-3 line answers to your question:


1) They allow you to treat various algebraic problems uniformly. For example, Commutative, Associative, and Lie algebras all have their own cohomology theories (Harrison, Hochschild, and Chevalley-Eilenberg respectively). These can all be seen as instances of a single operad cohomology.


2) One can use operads to construct cohomology classes for the Mapping Class group and Out(Fn) (and others). The idea is to use graphs "colored" by operads, and construct a chain complex out of these colored graphs that computes the desired cohomology. 


and perhaps the most classic answer:


3) They allow you to classify loop spaces and infinite loop spaces. For connected spaces, these are exactly classified as algebras over various operads. 


