Climate change communication on social media increasingly employs microtargeting strategies to effectively reach and influence specific demographic groups. This study presents a post-hoc analysis of microtargeting practices within climate campaigns by leveraging large language models (LLMs) to examine Facebook advertisements. Our analysis focuses on two key aspects: demographic targeting and fairness. We evaluate the ability of LLMs to accurately predict the intended demographic targets, such as gender and age group, achieving an overall accuracy of 88.55%. Furthermore, we instruct the LLMs to generate explanations for their classifications, providing transparent reasoning behind each decision. These explanations reveal the specific thematic elements used to engage different demographic segments, highlighting distinct strategies tailored to various audiences. Our findings show that young adults are primarily targeted through messages emphasizing activism and environmental consciousness, while women are engaged through themes related to caregiving roles and social advocacy. In addition to evaluating the effectiveness of LLMs in detecting microtargeted messaging, we conduct a comprehensive fairness analysis to identify potential biases in model predictions. Our findings indicate that while LLMs perform well overall, certain biases exist, particularly in the classification of senior citizens and male audiences. By showcasing the efficacy of LLMs in dissecting and explaining targeted communication strategies and by highlighting fairness concerns, this study provides a valuable framework for future research aimed at enhancing transparency, accountability, and inclusivity in social media-driven climate campaigns.