If the assumptions of the Black-Scholes model held, the surface would be flat. In actual markets, the value differs by strike and expiry, and its shape changes over time. Market prices are visible only for some strikes, so the surface is built by interpolating between those points.
A swaption has one more axis besides strike and expiry: the term (tenor) of the underlying interest rate swap. The values arranged along these three axes are called a volatility cube.