In my own words, a crude word to use in place of "state" is "memory". A state adds memory to a system in such as way that the output at a given time depends not only on its current input, but also on its previous inputs. There is also a more formal explanation about states here:
Discrete states can be thought purely as internal memory - for example a Unit Delay block has one discrete state, and it's output is computed based on two methods: Outputs and Update, which may be written as follows (u=input, x=state, y=output):
y = x;
x = u;
Here, the Simulink Engine never reads the discrete states, it only manages the memory allocation/de-allocation for the states. The delay is caused due to the fact that Update() runs after Output(), which means that the current "u" is only read into "y" at the next time-step when Output() runs again.
With continuous states however, Simulink asks the block to provide a derivative (dx/dt) of the state in the Derivatives() method and uses its ODE solver to compute the integral of dx/dt to obtain 'x'. This 'x' can then be accessed in the Outputs() function. For example, to implement an Integrator block, we might write:
y = x;
x_dot = u;
Here, the Simulink Engine reads x_dot and computes x for use in Outputs.
In both cases, at t=0, the value of 'x' is whatever what provided as the Initial Condition by the block.