Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism, their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a quantitative, causal understanding of why people approach activism. In this work, we develop a comprehensive causal model of how and why Reddit users engage with activist communities driving mass climate protests (mainly the 2019 Earth Strike, Fridays for Future, and Extinction Rebellion). Our framework, based on Stochastic Variational Inference applied to Bayesian Networks, learns the causal pathways over multiple time periods. Distinct from previous studies, our approach uses large-scale and fine-grained longitudinal data (2016 to 2022) to jointly model the roles of sociodemographic makeup, experience of extreme weather events, exposure to climate-related news, and social influence through online interactions. We find that among users interested in climate change, participation in online activist communities is indeed influenced by direct interactions with activists and largely by recent exposure to media coverage of climate protests. Among people aware of climate change, left-leaning people from lower socioeconomic backgrounds are particularly represented in online activist groups. Our findings offer empirical validation for theories of media influence and critical mass, and lay the foundations to inform interventions and future studies to foster public participation in collective action.