It works if you assume that the units are arbitrary. Suppose instead of people you're measuring heights. If there's going to be a distribution of leading number

It works if you assume that the units are arbitrary. Suppose instead of people you're measuring heights. If there's going to be a distribution of leading numbers in random data, it has to be the same in feet and in yards. To convert from yards to feet, you multiply by 3. Anything starting with 4 or 5 will always start with 1 after this multiplication, as will lots of things starting with 3 or 6. So starting with a 1 has to be as likely as starting with 3, 4, 5, 6. The same sort of thing has to work if you multiply by any number. Benford's law gives you the distribution where this works. The fact that this is scale-invariant means that it's not about preferring smaller numbers of things. It's just that the number of items in the range 10,000-19,999 is as large as the number of items in the range 0-9,999. Slightly bigger things also tend to start with ones. This all just assumes that the distribution of leading digits doesn't change if you scale your data, which I find intuitive enough to be satisfying.