Warren has a little proof of Benford's law in Hacker's Delight. To quote:
Let $f(x)$ for $1 \leq x < 10$ be the probability density function for the leading digits of the set of numbers with units. $f(x)$ has the property that: $$\int_a^b f(x) dx$$ is the proportion of numbers that have leading digits ranging from a to b. For a small increment $\Delta x$ in x, f must satisfy (Ed: I leave out a figure that shows the same graphically) $$f(1) * \Delta x = f(x) * x \Delta x$$
because $f(1) * \Delta x$ is, approximately, the proportion of numbers ranging from 1 to 1 + $\Delta x$ (ignoring a multiplier of a power of 10), and $f(x) * x \Delta x$ is the approximate proportion of numbers ranging from x to $x + x\Delta x$. Because the latter set is the first multiplied by x, their proportions must be equal.
The rest of it is very straight-forward, but I don't understand the bolded sentence which is the justification for the whole approach. It is clearly not true in general, so it must follow from the fact that we deal with numbers with units.
The assumption underlying Benford's law is that since the physical units are arbitrary, the distribution of physical constants should be invariant under multiplication by constant, which corresponds to a change of units (though no such distribution can really exist).
Another way of stating this is that the distribution of the logarithm of a physical constant is invariant under addition. In particular, if we look at the distribution of the logarithm modulo some real $x$, then it should be uniform in $[0,x)$. In particular, the fractional part of the $\log_{10}$ of a constant should be uniform in $[0,1)$. The leading decimal digit is $d$ if this fractional part is in the range $[\log_{10} 0.d,\log_{10} 0.(d+1))$, and since the distribution is uniform, this probability is $$ \log_{10} 0.(d+1) - \log_{10} 0.d = \log_{10} \frac{d+1}{d}. $$ This is Benford's law.