I have started studying control theory, and I have seen that state space representation is broadly used. However, it would be wonderful if somebody could explain me why it is useful and why it is so widely used.
$\dot x = A \cdot x + B \cdot u$
$y = C \cdot x + D \cdot u$
Here is a partial (and subjective) answer: Computers.
Initially, think 1800's the study of control theory was limited to classical analysis techniques. In general, strong results were only available for linear systems via techniques like study of transfer function properties, Routh-Hurwitz method, root locus, Bode-plots, circle criterion, Nyquist criterion etc. We worked a lot in the frequency domain because it was easier, and much more amenable to the tools at hand. Until up to the early 1930's this was the prevalent route for studying control systems.
But beginning 1940's we started having computational methods via computers. Then it made sense to put the systems into state space (essentially matrix forms) because computers were good with numbers but bad with symbolic computation. Then concepts from linear algebra, like rank of a matrix, invariant subspaces, null-space, kernels etc. found application in describing various properties of control systems like controllability, observability, feedback stabilizability etc.
Whats more; we could automate the process of finding these properties for the systems even when the dimensions made human computations prohibitive. It also allowed concepts like estimators, observers etc. to be developed which was not easy to do from a frequency domain perspective.
I think there are many many reasons why one would find state-space representations useful, but this is my answer to why they came to be so widely adopted.