How is the degaussing feature in the HMC5883L 3-Axis Digital Compass chip correctly used?
Why it's called degaussing...

Degaussing is traditionally thought of as a demagnetizing procedure. It's apt here, but you have to think of that magnetization in terms of its super-posed components. 

Consider this magnet in a magnetic field:



This is what the sensor elements in the earth (or other external field) experience. The "measurement" is actually the deflection of the needle from a known reference point (the center of the scale in this example) to a point on the scale created by the strength of the external field(s).

If a constant (or unequal) external field is applied over time, eventually the "spring" will lose some restoring force (tension). If you then remove the external force, the needle won't return all the way back to center.

In the magneto-resistive device case, the equivalent is a shift in the magnetic alignment vector. The domain of the sub-atomic particles is aligned with the origin field and when you add an additional field the particles are oriented away from this domain altering the electrical resistance of the device.

However, like the mechanical spring, the long term consistent exposure results in a gradual realignment of the origin domain and sensor stops being able to detect the external field as being different from it's reference "original" one.

To fix this, degaussing coils are present to provide a strong local impulse (momentary force) to remove any of these external influences and reset the magnetic domain to its original alignment, thus recalibrating the transducer.

So in a sense it is demagnetizing the device... by re-magnetizing it to the original orientation and removing the excess magnetization.

Usage in Magnetoresistive applications

For any specific device, you can find the actual activation procedure for the degauss coils in the datasheet. Typically it consists of two lines that are driven in opposition to run current in one direction through the coil, then in the reverse direction by flipping the polarity of the two signals. There is often an internal amplifier in the part between the control pin(s) and the actual coil(s) so that the inputs may safely remain high impedance and not pull too much current from whatever is driving the control pins (typ. microcontroller).

The harder questions are when and how frequently to perform this procedure -- that's the core of the question and the source of the OP's frustration.

Datasheets don't include answers to these questions because they are application-specific. Degauss operations have a high energy and interference (EMC) cost. You need to degauss when your sensor loses enough performance that it impacts your application. 

Performance is the combination of sensitivity and tolerance. The rate at which the sensor drifts (loses sensitivity or adds offset) is a function of the environment (how strong and unbalanced are the external fields). The amount of drift (error) that you consider tolerable is a function of your application (how deep is your ADC noise floor, how error-tolerant are your algorithms, etc).

If you don't have an external event available to use as a warning sign of reduced performance, the common practice is to simply reset at regular intervals.