With the increasing prosperity of the global economy, the protection and sustainable development of intangible cultural heritage (ICH) are being confronted with many problems, among which the difficulty of passing on ICH, particularly traditional handicraft, from masters to apprentices should be given attention. The main purpose of the study is, from the perspective of knowledge transfer and incentive mechanisms, to explore how to mobilize the positive factors to promote the successful implementation of passing on ICH. On the basis of identifying the key influencing factors of ICH knowledge transfer and establishing the payoff matrix between masters and apprentices, this study uses replicator dynamic equations to analyze strategy choices in different situations and verifies the impacts of these factors on passing on ICH through data simulation. The study draws the following main conclusions: (1) Increases in variables e, a, v, Ju, Jd, Ku, and Kd will drive up the probabilities of adopting a positive strategy and successful ICH inheritance and increases in variables Cu and Cd will drive probabilities down. (2) Changes in any variable have an impact on both sides through the interaction between masters and apprentices, and the impact’s direction is the same. (3) Among all economic measures, cost control should be the first to be considered. These conclusions provide important theoretical guidance for local government and related organizations to support ICH bequeathal and for masters and apprentices to make choices regarding strategy. The paper fills the literature gaps in the study of the influencing factors of ICH knowledge transfer and the game model between masters and apprentices, and on the incentive mechanism and strategy choice of passing on and developing ICH.