Fractional delay differential equations (FDDEs) have extensive applications in diverse fields such as physics and biology. The spectral delay correction (SDC) method for FDDEs was proposed as a solution, and a double-grid-based Legendre delay correction spectral method was developed. The double-grid technique was introduced to optimize the time and space discretization, while the Legendre polynomial was used for spectral delay correction, significantly enhancing the solution accuracy. A detailed error analysis was conducted through prediction and correction steps. The prediction step provided an initial estimate for the solution through an approximation, while the correction step further refined the approximation, thereby substantially improving the overall numerical accuracy. Numerical experiments demonstrate that the double-grid Legendre delay correction spectral method performed exceptionally well in solving FDDEs, greatly improving accuracy and fully verifying the correctness of the theoretical results.