Lets observe the U.S. ex post real interest rates from 1961 to 1986 :
At first look it is not easy to identify different states of the economy.
Now lets apply an econometric model that will try to identify those different states and observe the result:
The econometric model identifies 3 different states of the economy, this is what we call three regimes.
Basically the model firstly needs to determine the number of regime switch ( when the "state of the world" changes) : 2 regimes switch in this example.
Secondly once a regime switch has been identified, one or several parameters of the model will change.
In the above figures, two very basics switching models are employed.
One is a single intercept + some random errors.
Another is the same but with the variance of the errors that change at each new regime.
Regime-switching models that are employed are obviously much advanced, they have lot of parameters but the intuition is the same, models needs to detect structural changes in the series then some parameters of the models will be impacted by those changes.
So the intuition behind these models is to get a model with some coefficients that are regime
Ps: Illustrations are taken from the official documentation of PcGive (econometric software).