Vega tells you the effect of a one-point change in implied-volatility. Vomma tells you whether that sensitivity itself grows or shrinks as volatility rises. Out-of-the-money options have high positive vomma, which is why they gain disproportionately in a volatility shock.
Positive vomma is convexity in the volatility dimension: you make more on the way up than you lose on the way down. It is the reason far out-of-the-money puts are so persistently expensive in implied-volatility terms, and why selling them is profitable most of the time and catastrophic occasionally.
Example: two XYZ positions both have $500 of vega. The at-the-money straddle is nearly vomma flat. The $40 put has strong positive vomma, so in a jump from 20% to 45% volatility it gains far more than the straddle, even though they started with identical vega.
Related: vega-convexity, second-order-greeks, vanna, volatility-skew