In the wake of the COVID-19 pandemic, vaccine hesitancy has emerged as a critical global issue, including in India, driven by concerns such as political factors, vaccine efficacy, and potential side effects. This paper aims to provide a deeper understanding of the key drivers behind anti-vaccination (anti-vax) sentiment by categorizing opinions expressed on social media platforms, particularly Twitter. By employing advanced natural language processing (NLP) and deep learning techniques, we implement a multi-label classification approach to organize anti-vax tweets into 12 distinct categories, each representing a specific reason for hesitancy. The core of our methodology is a Nested LSTM (NLSTM) model, a novel multi-level memory RNN architecture that enhances traditional LSTMs by nesting layers rather than stacking them. The proposed architecture is designed to comprehensively capture and analyze the complex landscape of vaccine hesitancy, offering nuanced insights into the underlying concerns. We benchmark our model against state-of-the-art methods using various text representations, demonstrating a 12% improvement in macro F-measure compared to existing approaches.