Hmm. It could be that Apple's Archive Utility is to blame somehow when it opens the botanicula.zip file. So, you could download StuffIt Expander for free:
http://www.stuffit.com/mac-expander-download.html and use it to open the zip file instead of Archive Utility. Of course, you'll have to download the zip file again if you didn't keep it.
StuffIt's site will initially ask you for an e-mail address so you can "be added to their e-mail list," but you can give them a made-up e-mail address to get to the next step to actually download the application. Then use StuffIt Expander to open the botanicula.zip file...easiest way is to Control-click on the botanicula.zip file and choose Open With and choose StuffIt Expander. (The last step, of course, is to double-click on the Botanicula.app that was decompressed from the zip file. Hopefully, it'll work this time.)
If that doesn't work, I see that stooovie had a similar problem and was able to solve it:
http://forum.amanita-design.net/index.php?topic=3786.msg11115#msg11115Though his solution seems a bit overly technical (for me, that is), but I dunno; it worked though.
Also, another permissions solution that's not as technical: If you Control-click on the Botanicula.app and choose Get info, at the bottom it'll say Sharing & Permissions. Assuming you are the administrator of the computer, is your name listed there at the bottom? If not, and you are the administrator, try adding it. Choose Read & Write.