Is there no plugin you can use for this?
https://wordpress.org/plugins/
You can use PHP or Javascript (or others). With PHP you need to name each checkbox with a unique identifier so that when the user POSTs the data (presses submit) it can be identified and retrieved using PHP's $_POST Superglobal.
It's difficult to help here as the site doesn't allow code block and mucks up the code
If you read this answer:
http://stackoverflow.com/a/1010970
The "food[]" creates an array with the data for every checkbox that is ticked and submitted, and the value could be the domain name which you echo out on the next page or save to a file.
In your case instead of
name="food[]" you'd have something like
name="domain[]". then on another page which you POST to (using PHP POST method) you can loop the array $_POST['domain'] and echo out each domain they chose (Or use "print_r()", depends on what you want). You could use GET (url parameters) instead of POST if preferred.
Your list is not in a form either, you'd need to wrap the list in a form and define an action (POST, or GET, and the page you want to send the user to). Then you'd just make your form submit to whatever page or PHP template file you want, and output the POSTed data.
(Additionally, just above your list it states "get em cheap before there gone" - it should be "they're"

)