Why is Vega highest at the money?

Could you elaborate on why Vega, a measure of sensitivity to changes in volatility, tends to be highest when a financial instrument or derivative contract is "at the money?" This observation seems counterintuitive, as one might expect Vega to be highest when the contract is deeply in or out of the money. However, I'm curious to understand the underlying logic. Could you explain how the pricing dynamics and risk exposure of at-the-money contracts contribute to Vega's peak? Additionally, is there a specific mathematical or theoretical basis for this phenomenon, or is it merely an empirical observation? Thank you for your insight.