Well, that is very simple. Just sign up with your Twitter account. Click the your small account icon (top right) and select the menu item “Settings and privacy“. On the next page scroll to the end and click the button called “Request your archive“. A short information message appears.
Now you have to wait – usually a few minutes – for the “It’s Tweet archive time” email from Twitter. Follow the download button. Back on the Twitter page you can now download your archive.
In some cases – usually if you have a whole bunch of 200K of tweets or more – it could take a while until you receive the Twitter email containing the download link. As Twitter mentioned: Please be patient.