What is the computational universe proposition?
The basic premise is that below the dimensional limits of the quantum mechanical scale, the universe is discrete and everything that we perceive within it, such as space, time, matter and energy, are generated by a computational process. The fundamental components of our universe would then become data, the computational rules acting on that data, and the mechanism that mediates the computation.
The reason that we see physics as we do is because physics is a model of the way that we interact with the universe, and also the mathematics of physics is based primarily upon continuous functions. If the premise is correct then our physics is a ‘continuum’ based approximation to the discrete computations that occur at the deepest substrate of the ‘external’ universe.
Why is it Important?
Because if we could show that this proposition is valid then it would change the way that we view the structure of the universe. Quantum mechanics suggests that there is a limit to our ability to comprehend the fundamental structure of the universe. A computational based theory would enable us to comprehend the structures of the universe at a far deeper level. Physics cannot tell us what space, time matter, energy and many other fundamental constructs actually are. But the ‘reality’ of these concepts can be understood in terms of a computational model.
Does this mean that we live inside a computer?
I have to say yes to this because anything that computes can be termed a computer. However this does not imply that this computational machine was designed by other intelligent beings in another universe. It could be that the fundamental structure of the universe is discrete and computational just because it is, and for us to ask why it is this way has no meaning.
Where did the idea come from?
It was a guy called Konrad Zuse who was a pioneer in the development of programmable digital computers. Back in 1969 he produced a book called, ‘Rechnender Raum’, or in translation, ‘calculating space’. In his book Zuse proposed that our universe was the result of some sort of discrete computation, such as that from a cellular Automata. This idea has since flourished due to developments in several different disciplines:

Many quantum physicists started to consider that something may lie below quantum mechanics, and that it could be discrete.

The development of cellular automata theory by John Von Neumann and Stanislaw Ulam.

The discovery of complex and chaotic behaviour and its importance in physics; resulting in the understanding that very simple systems can generate very complex behaviour.

The development of powerful computers that could run complex iterative computations that highlighted this complex behaviour and enabled research into these ideas.
Who are the main supporters of this idea?
There are two scientists with significant reputations who support this premise and actively work in this area.

Professor Edward Fredkin of MIT who developed the discrete universe principle and has developed a basic approach to physics called, ‘digital mechanics’ that is based upon a computational paradigm

Stephen Wolfram used to be a physicist and is now one of the leading experts on cellular automata and supports this proposition.