Usability.
Your average desktop user does not want to go, 'Oh, well, I'm running on Processor X, with distribution Y, patch Z. I guess that means I need /this/ tarball (or this subdirectory of the big tarball).' Fat binaries solve this problem.
If I am a Mac OS X developer, fat binaries mean I don't have to make a separate Intel download, or separate PowerPC download. No worries about Joe User downloading the PowerPC version, then complaining about performance (not realizing they're running a PowerPC binary in Rosetta on an Intel machine), or so on. I can just have one download on my website, and the loader handles finding the correct binary.
Similarly, I can bundle 32-bit and 64-bit binaries for a given architecture into the same binary, rather than having separate 32 and 64-bit downloads (as is common on Windows). Tech-literate users may well know whether their system is 32-bit or 64-bit, but if I sat my father down in front of a brand-new Windows 7 machine from Best Buy, I doubt he would know whether to pick a '32-bit' or '64-bit' download for an antivirus program on a given website. He would, instead, call me.
Now, some software solves this problem by having a tiny installer you download, which then goes out and pulls down the correct packages from the Internet after examining your machine. This is one solution, though not entirely ideal (it means in order to do any install, you need to have internet access). Some installers include the entire set of binaries, and just install the correct one; this is fine, as long as you have an installer, but can break down if you try to transplant the hard drive into a new machine. For instance, Joe User picks up that nice Windows 7 Home Premium machine he saw at Best Buy, and plugs his Windows Vista drive in to copy over applications, unaware his old computer was running Vista x64, while his new Windows 7 machine is 32-bit. Joe has some Problems now, when he tries to run some of his old installed software that was 64-bit only.
At any rate, there are plenty of solutions to this problem; fat binaries are just one. None are perfect and all have their tradeoffs; in the case of fat binaries, the main problem is disk space. Package management tools have their own problems. (RPM dependency hell any time you want to go outside of your distribution's available packages, for instance, and the 'screw this, I'm installing PHP from source' result some sysadmins turn to.)
From a server standpoint, fat binaries aren't necessarily the most useful solution (unless you're dealing with clustered machines with variant processors or configurations, but a shared filesystem between them), but from a *desktop user standpoint*, fat binaries may be friendlier than other options.
At any rate, my *personal* opinion is that from a general desktop end-user standpoint (as opposed to a sysadmin/techy standpoint), disk space is cheap but usability is priceless. And my experience is that fat binaries require less work on the part of the end user (though, admittedly, more work on the part of the developer; building Universal Mac OS X binaries of software outside of Xcode can be a hair-pulling experience at times and inspire fond thoughts of Windows installers that just pick the right binary based on a system check).
So whether you feel Linux benefits from fat binaries may well boil down to whether you feel Linux needs to target general, non-techy desktop users more or not. Your own opinions may well differ from mine; not everyone's criteria and priorities are identical, which is probably a good thing. Otherwise we'd have a pretty homogenous software community out there!