OK, try once again but also make sure you are using the small "i" button (see description below) to download, as this seems to reduce the likelihood of getting a corrupted download.
To download and install games through Origin
Download and install the Origin client for Mac and make sure it is up-to-date. If you already have Origin on your Mac but are having problems with downloading or installing games, delete the client and download a new copy (this sometimes fixes issues with downloading). The latest version can be downloaded from here: http://www.dm.origin.com/mac/download
Additional advice on how to download games (thank you Tremayne for this tip):
When downloading from Origin, do NOT use the first install icon. Move your cursor over the game cover and click on the information (letter i) and then click on the install button from there. For some reason, downloads from Origin frequently become corrupted. Using the i reduces the chance of your download being corrupted. If you have already been unfortunate and have a corrupted download, then it is important to remove all traces of this before trying again, by doing a thorough, clean uninstallation first.
Are you running anti-virus software or any clean-up applications? If so, disable these, as they might be deleting files that Origin needs. A fast, stable internet connection will also help you.