Officially, the only way to play the Linux version is with the direct-download purchased from 2dboy.com
However the game files in the Linux version are the same as under Windows. So if you have purchased the Windows version in a box or from Steam, you can install the Linux version by upgrading the demo.
Simply install the Linux demo, and then copy over the "res" directory from your Windows installation.
Soultaker wrote the following on the 2D Boy forum:
Officially, the Linux version is only available to customers of 2D Boy, but unofficially, if you own a copy of the Windows version, you can upgrade the Linux demo of the game to a full version as follows:
1. Install the Windows version of the game somewhere (this could even be on Ubuntu using Wine); this usually goes into C:\Program Files\2D Boy\World of Goo.
2. Install the Linux demo of the game (which goes into /opt/WorldOfGooDemo).
3. Copy all the files in the res/ subdirectory of the Windows installation to the res/ subdirectory of the Linux installation, overwriting existing files.
Of course, installing the full Linux version with the official package is easier. Also consider that 2D Boy makes less money from sales through third-party resellers (Amazon, Steam, et cetera) and that they developed the Linux version at their own expense after the Windows and Mac versions were already released, so the moral high road would probably be to buy the game from 2D Boy if you like to play the Linux version, but I leave that decision up to you. (I can imagine you don't want to return a gift for this reason.)