There is no easy way to do it. Right now your sole option is to visit the transcript(s) you want to download then save the pages themselves locally through your browser (File → Save As or whatever). You'll have to do this individually for every day you want to save.
The general form of the transcript URLs (that the View Transcripts link takes you to), if you ever want to visit one directly, is:
Where, taking the parts of that example:
chat.meta.stackexchange.com is the chat server (SE has a few).
89 is the chat room ID.
2017 is the year.
4 is the month.
6 is the day.
0-21 is the time of day to view the transcript for.
So one tiny trick you can use is for busy rooms where the transcript of a day is divided into time blocks, just use
0-24 on the end and you can view the whole day. That's about as much as you get in terms of shortcuts.
That said, incidentally, I wrote a utility to scrape all the transcripts of a given room recently, and plan on creating a little web application in the very near future to let folks browse chat transcripts and search chat more easily (and pull fun stats, like emoji leader boards and such). I'll try to remember to post back here when I do, or just ping me in The Tavern.
But, other than writing your own tool to scrape transcripts, no, there's no way to mass download transcripts aside from visiting each day and just saving the page (which might be just fine for your needs).
Note that there's also a Load to my last message button on the main chat window, so if you're trying to catch up, sometimes that button is helpful (but not always), and you can save the web page to your computer if you want to read it later.