These functions are very useful for example if you need to control when a visitor in your website place an order and you need to check if he/she didn't clicked the submit button twice or cancelled the submit just after have clicked the submit button.
If your visitor click the stop button just after have submitted it, your script may stop in the middle of the process of registering the products and do not finish the list, generating inconsistency in your database.
With the ignore_user_abort() function you can make your script finish everything fine and after you can check with register_shutdown_function() and connection_aborted() if the visitor cancelled the submission or lost his/her connection. If he/she did, you can set the order as not confirmed and when the visitor came back, you can present the old order again.
Note that if you use ob_start("callback_function") in the begin of your script, you can specify a callback function that will act like the shutdown function when our script ends and also will let you to work on the generated page before send it to the visitor.