Why does a 10-decibel increase represent a 10-fold increase in sound intensity?

The decibel scale is logarithmic, not linear, which is why a 10-decibel increase represents a 10-fold increase in sound intensity.12

The relationship is defined by the formula for sound intensity level ($b$) in decibels:
$$b=10{\log }_{10}\left(\frac{I}{I_0}\right)$$
In this equation, $I$ is the intensity of the sound being measured and $I_0$ is the reference intensity. Because the unit is based on the common logarithm (base 10), each increase of 10 decibels corresponds to multiplying the intensity ratio by a factor of 10.32

For example, if you increase a sound level by 10 decibels, the ratio of the intensities ($I/I_0$) increases by a power of 10. This logarithmic scale is used because it effectively compresses the vast range of sound intensities that the human ear can perceive into a more manageable numerical range.21