Cookies are generally harmless bits of data websites store on your own computer (in the browser) so that it can do things such as refer to you by name when you return to the site, quickly show you things you have previously ordered, remember your birthday, the number of times you have visited, and so on.
Not all cookies are quite that innocuous, but most are — and they are generally not a security issue. That said, they can fairly be seen as a potential privacy issue, so if you want to clear them you can generally do so from the settings of any browser you use.
Regarding shownotes.ca specifically (Alan’s logbook of items from his radio shows), if you have that bookmarked from before July, you should delete the bookmark, visit shownotes.ca again by typing it directly into your browser’s address bar, and then re-bookmark it.