What happens depends on if you have a put calendar or a call calendar. Say you have a 50 strike calendar that you let the short leg expire in the money:
With a put calendar, you will be assigned 100 shares at a price of 50.
With a call calendar, you wind up with a short sale of 100 shares at a price of 50, or, if you happen to own 100 shares of the stock, you would wind up selling the stock at a price of 50.
In both cases, you keep the entire up-front credit that was collected. Also, the long legs of your calendar remain intact since those options did not expire.