I understand their abstract definition but having trouble applying the HMM method to Forex prices. What should the observations be? Then what should the states be (like "hot", "cold", etc.)?
Your decision. You can define the states to be "positive return" or "negative return". So if the return is negative, then its state would be that "negative return".
Or, you could even model them with continuous emission so as to define a state with a specific mu and variance. So that given the return at time T, you can say "it could technically be possible that this return be in any one of these states, but it's most likely that it's at this state given its mu and variance"