. . . Still not getting what exactly expires though... 
Imagine that you were running a business and accepting Bitcoin payments with Electrum.
Imagine that you use Bitcoin Addresses like invoice numbers, so everytime a customer owes you a payment, you issue a new bitcoin address. This allows you to keep track of which customers have paid, when they paid, how much they paid, and what the purpose of that payment was.
Imagine that you'd like to be able to keep track of which customers have paid you in a timely manner, and which customers are overdue.
You can set an expiration on the payment request. Then, by looking at the list of payment requests, you can see who still has plenty of time to pay, whose request is about to expire, and who is overdue and needs to be sent a reminder with a new payment request (probably with a new payment amount since they now have late fees).
If a payment is sent to an address that is associated with an expired request, you'll still receive the payment to Electrum, but you'll know that the payment came late and that you need to contact that customer and let them know that they owe you a late fee due to their un-timely payment.
