Together with length in inches it gives the size: a "5 × 50" is five inches long and 50/64 of an inch across. Dividing the ring gauge by 64 gives inches; multiplying it by 0.397 gives millimetres.
Does ring gauge change the smoke?
It changes the proportions. A wider cigar carries more filler for the same single layer of wrapper, so the wrapper contributes proportionally less of what reaches you — which is why a blend often tastes different across sizes in the same line, without anything about the blend having changed. Diameter also affects how hot a cigar burns and how much resistance it offers.
Why they got wider
The average ring gauge sold has risen substantially over recent decades. That is a shift in what the market buys rather than any technical development, and sizes that were once unusual are now ordinary.
The vocabulary is not standardised
Ring gauge is an exact measurement, but the commercial name attached to it is not. Two houses will sell noticeably different cigars under the same shape name.