Q: How do i re-download apps from the iphone to my mac?

Which section have you tried, going to the Purchased section in the iTunes store, selecting Apps and clicking on the cloud icon on an app to redownload it, and/or connecting your phone to your computer's iTunes and using the File > Devices > Transfer Purchases menu option to copy them over ? What happens when you do that ?

The iTunes account(s) that downloaded the apps on your phone is authorised on that computer's iTunes ? And do the apps show in the Purchased section in iTunes for redownloading e.g.

If you try downloading them from there does that work ?

And that is the computer that the phone is normally synced to ? Have any other purchases (e.g. music, films etc) that are on the phone copied over via Transfer Purchases ?