You can register the codes for the games that you have on discs with Origin, then download the whole lot through Origin. There is no need for discs or a disc drive if you do this.
The first thing to check is that your Mac is sufficiently up-to-date to be able to run the Origin client. It requires at least OS X 10.6.8 with an Intel Core 2 Duo processor or better.
To download and install games through Origin:
First install the Origin client for Mac and make sure it is up-to-date. 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.
Caspin