This is quite technical writing about programming concepts; words such as state have specialised meaning.
Loosely speaking, state is the value of some variables at a particular moment.
The problem here is that when doing some asynchronous work the initiator of the work tells the worker
Do some work, when you have finished please call this callback function with the result
The initiator then proceeds to do other processing, it never directly "hears" the result from the worker. However the callback function will eventually get called with the worker's results, and at that point may need access to state held by the initiating process.
Very Contrived Example:
Initiator: customer X wants to buy 3 units of stuff. At this point the initiator has state such as information about the customer, and the order.
Initiator needs the price of 3 units of stuff so calls the Pricing Engine, telling it to call the PrepareInvoice function when the price is determined.
PrepareInvoice Callback function receives the price from the Pricing Engine (simply passed from the Engine as it calls the function) BUT needs the Customer information, which is state known only to the Initiator.
There are idioms in many Programming languages to allow us to solve this problem.