How to intuitively understand the state space formulation of discrete time system?
Let me start by answering your last question


  How can this model be used to describe the state occurring at all times?


If you know the state at one single time instant $k$ (e.g. by a given initial condition when the system starts operating), then, given an input signal for times $\ge k$, you can use the state-space equations to compute the output signal for $k+1$, $k+2$, etc. to infinity. The state at a time $k$ contains all information about the past history of the system necessary to compute its current and future output given the input signal.

As for the other question why the state equations only relate time $k+1$ to time $k$ and not also to times $k-1$, $k-2$, etc., the answer is again that the state summarizes all past history of the system. Note that in general the state $x$ is a vector. E.g. if you have a discrete-time linear time-invariant system specified by a linear difference equation with constant coefficients, the output is given by a linear combination of past output values, and current and past input values. Such a system can be implemented by a tapped delay line as shown in the figure below for a second-order system. Now if you define the state vector $x$ to be the content of all delay elements at a given time, you kind of exchange the explicit formulation of time dependency spanning several past time indices with a vector-valued equation spanning only one time interval. The more memory a system has, the higher will be the dimension of the state vector. So by using a vector difference equation (i.e. a system of equations) instead of a scalar equation, you can always obtain a first order equation relating time $k+1$ to $k$ and to no other past time indices.


"Biquad filter DF-II" by Akilaa - Own work. Licensed under GFDL via Wikimedia Commons