Everything with a headphone jack already has a headphone amplifier. People are mostly talking about dedicated standalone headphone amplifiers, which may have different characteristics than the integrated headphone amplifier.
There are generally three main reasons people get headphone amplifiers:
1. To boost the signal, to make the headphone play louder. Some headphones are relatively quiet and need a higher signal (larger voltage) or just something that can handle the required current. A more powerful amplifier may be able to provide more power into different headphones.
2. To boost the sound quality. An amp pretty much takes an input signal and replicates it on the output (optionally scaled higher or lower, to increase or decrease the volume). As with any type of analog system, this can't be done 100% perfectly, so there are imperfections on the output as it does not track the input completely correctly. A cheap integrated amplifier may do a suboptimal job with this. Depending on the design of the amplifier circuit, the performance at different output volumes into different kinds of headphones with different input signals, may be audibly different. It depends. Some amplifiers intentionally distort the signal so the output has somewhat different characteristics than the input; in some situations, this type of distorted sound is pleasing and preferable to many listeners.
3. To get a shiny new toy (practical benefits aside), or because of peer pressure or misinformation.
Particularly for sensitive high-end IEMs there may be many factors behind #2, other ways that sound quality could be improved over something that's deficient. But in general, sound quality depends much more on the headphones than on the other components, which are much easier and cheaper to produce reasonably competently. edit: some amps also have controls for crossfeed or equalizer options like bass or treble boost/cut, so you're gaining extra functionality in some ways. These are things that can easily be done in software, but crossfeed is somewhat rare on portable music players.
If you're looking for better sound quality, you want to avoid the cheap pro headphone distribution amps/mixers, that are not designed for high fidelity listening but just for hearing your own sound or the mix in the studio. The ones you linked would fall into that category.
Edited by mikeaj - 2/11/12 at 9:48pm