Headphones may be insensitive. They may have a high impedance, or an impedance that varies a lot with frequency (most vary to some extent).
Headphone outputs can have problems driving headphones (to an adequate volume or with a flat frequency response) because of:
1 Inability to source sufficient current.
2 Inability to achieve sufficient voltage swing.
3 Too high an output impedance.
4 Any 2 or 3 of the above in combination.
Very rarely some headphones perform better with a high output impedance (some are designed to operate with 600 ohms out, an old standard), but an amplifier with good voltage and current capability can be made to appear as a high driving impedance by using a series resistor, whereas nothing can lower output impedance, or improve current or voltage performance other than an amplifier (OK, a transformer can, but let's not go there).
A headphone amplifier is not necessary in some instances and will not be an improvement in some instances, but often it will. It's the difficulty of defining and understanding the circumstances in which it will be an improvement that makes it a good recommendation in general, if the amplifier itself is of good quality (has good current and voltage and low output impedance in combination with low noise and distortion).
Put simply, it's easier to recommend trying a good amplifier than to explain to a naiive consumer with phones and a DAP of unknown quality how to figure out if he needs one, and owning one has the advantage of opening up the range of headphones he can expect to use successfully should he decide to buy different ones in future, so it's often chosen as an easy recommendation without any intent to mythologise.