Emergency services rely heavily on Twitter for early detection of crisis tasks to enhance crisis management systems. However, employing state-of-the-art models often face data sparsity as well as their inadequacy to handle long-range dependencies between tweet tokens. Additionally, the authorities need to gain confidence in the model's prediction so that the detected task information can be better believed and prioritized. In this study, we present a generalized framework named Explainable Attentive model for Crisis Task identification (ExACT) to handle the above mentioned challenges while identifying crisis task relevant tweets as well as provide the model explainability by utilizing a very small corpus of tweets. The novelty of ExACT is three-fold: (1) Data enrichment has been introduced by non-dynamic contextual attributes derived from tweets to overcome the sparsity. (2) Feature enrichment has been incorporated using hierarchical attention at both local and global levels using residual self-attention and correlation attention to capture long-range dependencies. (3) LIME based model explainability approach added to understand the task important tokens. Experiments reveal that ExACT has a competitive performance improvement over various state-of-the-art models in terms of F-measure (ππ% and ππ% respectively) and accuracy (ππ% and ππ% respectively) across two different crisis tasks infrastructure damage and support signal identification. Consistent performance improvement for two different tasks considered from publicly available crisis event datasets depicts the model's generalizability. While LIME supported explainable mechanism in ExACT can identify the important keywords but does not guarantee a high score in terms of plausibility and faithfulness metrics.