This question is a little bit technical to answer because it depends on many different factors, especially the settings of your web browser.
This is what I think happens usually: when you log in to iTalki, your web browser sets a 'cookie' to show that you are online. This 'cookie' persists until you log out again, or until you quit the browser program. When you click the link to an external web page, what happens depends on if/how you are using 'tabbed browsing'. If the link opens in another tab, you will remain logged in on the first tab and will be able to carry on in that tab as if nothing had changed. If you are not using tabs, or if the link opens in the same browser window, you will still be logged into iTalki even though you cannot see it. You can test this by using the 'back' navigation button or by typing www.italki.com into the browser's address bar - if you do not have to log in again, then you are still logged in.