Why is it said that standing waves do not transfer energy?

Standing waves are always the result of the interference of two (or more) waves. E.g. in a rope or a string: the interference between a wave and the reflected wave (when the first wave reaches the end of the rope, it reflects) travelling in the opposite direction, as can be seen below. It is interesting to notice, however, that both these waves have the same frequency.

^{Image source: Wikipedia.}

If this situation appears in an infinite rope, standing waves always are the result. In finite ropes (which are more common in musical instruments and the reality in general), standing waves only occur if there is a clear relationship between the length of the rope and the wavelength.

The reason it is said that no energy is transferred, is seen in looking at a single point on the rope. The energy stored in the vibration is $\frac{1}{2}m\omega^2 A^2$. Since the frequency of the vibration ($\omega = 2\pi F$) and the amplitude ($A$) is constant for each individual point in a standing wave, the energy for every point remains constant.

As standing waves only occur if the frequency of the two travelling waves are equal and if certain conditions are met (relation between length of rope and wavelength), the fact that the two waves travel in opposite direction is not good enough.

So standing waves result in a violin string from moving a bow across the string. As explained above, the energy in a single point of the string in a standing wave is constant. Or it would be, if energy would not be lost (transferred) to the air around the string due to air resistance

Instruments can be heard because there are leaks of energy: the "standing" wave would'nt stand infinitely (without new input of energy). For cords intruments, a bit of energy diffuse via friction with air, but most diffuses via the cords end then conduction to the main bed then large communication of motion to the air (then up to your ears).

But within the cord, there is no neat transport in one or the other direction. Your guitare is not pushed away (or is pushed in opposite directions at the same time) :-)