Benford's law will hold approximately for any set of numbers with the property that they are distributed over many orders of magnitude, from a distribution which doesn't change much if you multiply by a random number in some range.

An example of such a set of numbers is the set of numbers that come up in intermediate calculations involving a lot of different numbers. (This explains the logarithm books where the phenomena was first noticed.)

Another example are the numbers you see coming out of any sort of self-similar phenomena. As fractals show, self-similar behavior is ubiquitous. As a result numbers like the length of rivers, the height of hills, and the size of cities all tend to follow Benford's law.

For any particular source of numbers, the explanation for why they fall into a category that matches Benford's law will differ. Benford's law is a property that mathematical models tend to have, rather than being a rigorous mathematical theorem.

(FYI Benford's law is something that I've known about, and thought about off and on, for close to 20 years ago now.)