How does a whirling hygrometer work, how does it differ from static dry/wet bulb thermometer, and how to obtain humidity from its readings?
Instead of «whirling thermometer» you probably refer to «sling psychrometer» consisting of a dry-bulb thermometer and a wet-bulb thermometer.  Yet, different to a static wet-bulb thermometer shown in the question, the constant rotation of the two thermometers permits evaporation from the wet cloth around the web bulb thermometer to «pristine parts» of surrounding air.

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If the air around the wet-bulb thermometer is less than water saturated, the temperature of shown on the wet-bulb thermometer will be lower than the one read on the dry-bulb thermometer.  The swirling motion aims to expose the wet-bulb thermometer to air which was not yet in contact with the former.  Thus, both recordings allow you to report both (dry) temperature, dew temperature, and humidity of the air.
A beginner's introduction is equally given here, a computation chart relating the temperature difference read (one for the Celsius, an other for the Fahrenheit scale) to the relative humidity here.

Suggested by user @uhoh, it is worth to set the quality of the readings obtained with this instrument into a context.
The 1993 edition of Conserve O Gram 3/1 (accessed from the series' program overview) mentions «These instruments are accurate to $\pm \pu{2\% RH}$ when properly maintained and operated by trained and practiced users.»  Equally, the guide reminds to the Clausius-Clapeyron relation when stating:

«Psychrometers and psychrometric charts are intended for use within a certain range of atmospheric pressure. High altitude, above 900 m (approx. 3,000 ft), will directly affect the accuracy of the RH reading unless a pressure correction formula is applied or a psychrometric chart or slide rule for the appropriate pressure is used. For further information see Ann Hitchcock and Gordon C. Jacoby, “Measurement of Relative Humidity in Museums at High Altitude,” Studies in Conservation 25 (1980): pp. 78-86. (Copies are available to NPS sites from Curatorial Services Division, Harpers Ferry office.»

A study by Wang and Tang /  Singapore's metrology centre published by 2007 mentions:

«... that although the psychrometer coefficient of a particular psychrometer can be quite consistent with long-term stability typically within the range of  ± 1% rh equivalent, there are larger variations (up to 4% rh equivalent) among different psychrometers even of the same type.»