To alleviate the impact of fake news on our society, predicting the popularity of fake news posts on social media is a crucial problem worthy of study. However, most related studies on fake news emphasize detection only. In this paper, we focus on the issue of fake news influence prediction, i.e., inferring how popular a fake news post might become on social platforms. To achieve our goal, we propose a comprehensive framework, MUFFLE, which captures multi-modal dynamics by encoding the representation of news-related social networks, user characteristics, and content in text. The attention mechanism developed in the model can provide explainability for social or psychological analysis. To examine the effectiveness of MUFFLE, we conducted extensive experiments on real-world datasets. The experimental results show that our proposed method outperforms both state-of-the-art methods of popularity prediction and machine-based baselines in top-k NDCG and hit rate. Through the experiments, we also analyze the feature importance for predicting fake news influence via the explainability provided by MUFFLE.