The one thing to add to this is that usually each state doesn't emit a single token ("I am in state 1" then "I am in state 2") but instead you assume that each

The one thing to add to this is that usually each state doesn't emit a single token ("I am in state 1" then "I am in state 2") but instead you assume that each state has a range of possible actions and the likelihood of a choice of action varies with state. So if might not be that your model is sunny versus rainy but instead cold front v warm front. Since rain is more likely during a cold front your observation of rain increases your belief that the system is in the "cold front" state.