With the increasing penetration of renewable energy sources, there has been a profound structural transformation in electricity markets, particularly in the operational paradigm where demand-side resources participate in market mechanisms. This evolution underscores the critical role of demand-side flexibility in enhancing grid resilience and accelerating low-carbon transition pathways. The traditional supply-side regulation is gradually shifting towards a multi-agent decision-making model dominated by demand-side management, involving complex interactions among government, electricity suppliers, grid operators, and consumers. Evolutionary game theory (EGT), as an important tool for studying multi-agent dynamic games and strategy evolution, demonstrates unique advantages in analyzing demand-side participation in the electricity market. Unlike traditional game theory, EGT accounts for bounded rationality, incomplete information, and dynamic strategy adaptation, enabling effective analysis of market equilibrium and stability under different incentives and policy frameworks. A comprehensive review of the applications of EGT in the electricity market is provided in this paper, focusing on the strategy evolution and equilibrium mechanisms of demand-side participation. It explores how EGT helps to understand and predict strategy adjustments and the impact of policy incentives on market stability in multi-agent games. The paper also looks forward to future research directions. By organizing and analyzing the existing literature, this review offers a theoretical framework and practical reference for policymakers, market regulators, and researchers to facilitate the collaborative evolution and sustainable development of the electricity market under the context of renewable energy.