Thanks for sharing the error message straight away, that's a big help.
It says "read-only" so it's possible you might not have extracted the game properly before running it, or placed it in an incompatible location. What you will need to do is extract the grove Zip, and then place it in the applications folder:
Right click the file you downloaded, use the archive utility to unzip the files, then drag the grove app into your applications folder. Hope that'll work!