Does this actually work, though? I was under the impression that most phones sold in the US have incompatible GSM frequencies to those sold in Europe.
The parts that he's wrong would be those few "dumb" phones that companies still sell that aren't able to use the other frequencies, but this is rare. Also, some countries don't use the same "LTE" or "4g" frequencies that the US uses, so if you were to bring your phone there, it would still work, but it be limited to "3g" instead of "4g" speeds, but this varies country to country and it also depends on the network and plan that you choose.
A quad-band "world phone" (that supports 850/900/1800/1900) will work in _most_ places (some exceptions) in the world that support GSM. There are some countries that use some obscure bands - Benelux, Russia etc. use 450MHz.
It gets a little more complicated when you get to UMTS/HSDPA/HSUPA... but Wikipedia explains it all quite nicely http://en.wikipedia.org/wiki/GSM_frequency_bands
So your iPhone will work everywhere - at least everywhere that uses at least one of the 850/900/1800/1900 MHz frequencies... if it's unlocked.
On 2G, 2100MHz was only on CDMA, which most of the world (still) doesn't support... so if you've got a CDMA phone and are travelling anywhere outside of a very limited list of some 45 countries (North/Central/South America, Caribbean, Far East), you're SOL.
So like I said UMTS/HSDPA/HSUPA gets more complicated, because then you're not just talking GSM or CDMA... your phone needs to talk UMTS/HSDPA/HSUPA as well as having the correct frequencies and GSM or CDMA.