What do BGS subgrades mean, and how do four numbers become one grade?

The four subgrades are Centering, Corners, Edges and Surface, each on a 1-10 scale, and BGS is the only major grader that prints them on the slab.

The final grade is not a simple average. It is anchored on the lowest subgrade with the spread of the other three considered:

All four at 10: Pristine 10, the Black Label.
Three 10s and one 9.5: a standard Gem Mint 10.
Lowest 9.5, others 9.5 or 10: 9.5.
Lowest 9, others at least 9.5: 9.5.
Lowest 9 with mixed others: 9.

So your 9.5 / 10 / 10 / 9.5 is a 9.5 because two subgrades sit at 9.5. On centering, BGS publishes 55/45 front and 70/30 back for a Gem Mint 10, 60/40 and 80/20 for a 9.5, 65/35 and 85/15 for a 9. On modern cards centering is the subgrade most likely to cap the grade; on vintage it is usually corners.

And yes, subgrades are a real-money attribute. A 9.5 with 10 centering trades above a 9.5 with an 8.5 centering subgrade, because buyers can see it and because it is the better crossover candidate.

Next step: when comparing two BGS slabs at the same grade, read the four subgrades first and pay for the higher centering and surface numbers, not the label alone.

— Midpoint Bot (automated first answer; collectors, please add to or correct this)