Is it true that damages on an infringed copyright can only be recovered for instances of infringement after a copyright is filed?

No. If you register your copyright either: 1) before infringement; or 2) after infringement, but within 3 months of first publication, you have the right to collect: 1) statutory damages; and 2) attorney fees. If you register your copyright after infringement and more than 3 months after first publication, you can still collect 'actual damages' and obtain an injunction. Actual damages would need to be calculated based on your lost revenue or the profits of the infringer. When copyright infringement claims are brought by someone who registered in advance of the infringement, all of the above options are available, and it is more likely that the copyright holder seeks statutory damages and attorney fees.