Frank nicely explained the Windows-specific verification. I’ll give a more general answer.
What does Verify disc after burning actually do to verify the data?
I mean, does it create a hash of the source and destination content and then compares them? If it is so, does it store the hash of the burned content in RAM? Or does it save it in a temporary file on the hard drive? Is there a log file of what is going on?
That is certainly one way that a comparison can be implemented: hash one file (hopefully with a sufficiently large—read low chance of collision algorithm), repeat for the other, and compare hashes. If that is how a verification is implemented, then you will be able to see the drive-LED flash for a while, then the CD/DVD-LED flash for a while.
Another way to implement the verification is to read a block of one file, then the same block from the other file, compare them, then repeat until the end of the file is reached. In this case, you will see the LEDs of the two drives alternating back and forth.
Of course, if the the hard-drive and optical drive don't have LEDs, then it won’t be as obvious, but you can still see it with something like ProcessMonitor because it will log a series of reads from one, then the other either in a single, big burst each or alternating, small bursts.
I imagine it is some kind of comparison between the original files and the files that have been burned on the disc, but does anybody know how it is really done at a low level?
Actually, all it really does is to flush the drive cache so that the comparison function is reading the data from the actual disc instead of from the memory cache. Obviously this is a critical step because if the verification is done from cache, then it does not represent what is actually on the disc, so corruption can easily slip through.
You can see whether a comparison is done from the drive or from the cache in RAM by how fast it occurs. If you manually do a simple comparison (e.g., with WinDiff or WinMerge or by hashing them with a hashing tool), you will notice that the comparison happens much faster than expected because it is reading the files from memory cache. You must flush the cache to force it to read from the actual disc. For optical drives (and other removable media like flash-drives and memory-cards, simply ejecting the drive is enough to flush the cache, but for hard-drives, it is not nearly as simple (though usually that doesn’t matter because the new copy is the one you want to test).