How the website owner keep track of your times of access?
I am not sure how this website keeps track but I can tell you the general flow how usual websites keep track of user's data.

If the you are registered user on the site it doesn't matter which ip you use it can be tracked easily.
To be more specific when you read the complete article a tracker can be anywhere from page-end scroll to read-more button.

Edit1: Looks Like the website you are talking tracks user based on (article click+ip+ cookie) so it is easy just to choose new ip + incognito browsing to get 30 article limit each time.