You have to manually copy it off an save it somewhere on your own. Either locally to your system or in a blog/forum.
How to Save Computer Analysis on New Chess.com?
Finally there is a built-in feature in the new chess.com to do this - see the 6/9 news update for more info

Yes! Saving the analysis is a new feature. I've found it very useful for myself. For example, I can run the computer analysis on a game I just played. Then I can pick up the analysis by download later and store it in my personal games played database. Or, this really lets you effectively use Chess.com for your personal games database since keeping the analysis and looking it back up whenever you want is now an option.
On the NEW chess.com, when I run a Computer Analysis on a completed game (e.g., a Deep Analysis which runs for about 4 minutes), I can't figure out how to save the results of that analysis so that I can come back to it again to study it. Anyone know how to do this?