Existence at a place or point of time... i.e. how you are (or how something is).
This could also be rephrased as:
"However, only ‘used to’ is possible when we talk about things in the past."
Some more common examples of "state":
"There are four states of matter: solid, liquid, gas and plasma."
"He's in a confused state of mind."
"Your apartment is so messy! What a terrible state it's in!"