As the scale of power system continues to grow, a fast and accurate distributed optimal power flow solver becomes crucial for the effective dispatch of power system. This paper presents a learning to optimize (L2O) approach to accelerating the distributed optimal power flow solving. The final convergence values of global variables and Lagrange multipliers of the alternating direction method of multipliers (ADMM) are estimated as its warm-start solution. A long short-term memory-variational auto-encoder (LSTM-VAE) model is developed as the core for estimating the convergence value, and the LSTM-VAE assisted ADMM is proposed. The LSTM generates low-dimensional representations of global variables and Lagrange multipliers, while the decoder part of VAE reconstructs the high-dimensional asymptotic convergence values. A novel loss function is designed in the form of a quadratic sum penalty term to incorporate the constraint violations of the Lagrange multipliers. Additionally, a two-stage training data generation strategy is proposed to efficiently generate substantial data within a limited amount of time. The effectiveness of the LSTM-VAE assisted ADMM is evaluated using the modified IEEE 123-bus system, a synthetic 500-bus system, and a 793-bus system.