Sadly, you can't.
The only way of saving the chat text, in one go, is by viewing the presentation live and copying the chat just before the class ends. Copying the chat from the recording is difficult and takes a lot of time as you have to do it piece by piece up to the end of the video.
You may possibly find a copy of the chat recording, made by a facilitator, on the Weekly Webinar page in the "Chats from the Webinars" folder.
If you don't see it there, you may ask for a copy of a specific chat text in the relevant forum.
(Edited by Thomas Hodgers - original submission Monday, 24 March 2014, 12:54 PM)