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 Meta (previously known as 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. 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. Additionally, we conduct a comprehensive fairness analysis to uncover biases in model predictions. We assess disparities in accuracy and error rates across demographic groups using established fairness metrics such as Demographic Parity, Equal Opportunity, and Predictive Equality. Our findings indicate that while LLMs perform well overall, certain biases exist, particularly in the classification of male audiences. The analysis of thematic explanations uncovers recurring patterns in messaging strategies tailored to various demographic groups, while the fairness analysis underscores the need for more inclusive targeting methods. This study provides a valuable framework for future research aimed at enhancing transparency, accountability, and inclusivity in social media-driven climate campaigns.