Explainable artificial intelligence is increasingly employed to understand the decision-making process of deep learning models and create trustworthiness in their adoption. However, the explainability of Monocular Depth Estimation (MDE) remains largely unexplored despite its wide deployment in real-world applications. In this work, we study how to analyze MDE networks to map the input image to the predicted depth map. More in detail, we investigate well-established feature attribution methods, Saliency Maps, Integrated Gradients, and Attention Rollout on different computationally complex models for MDE: METER, a lightweight network, and PixelFormer, a deep network. We assess the quality of the generated visual explanations by selectively perturbing the most relevant and irrelevant pixels, as identified by the explainability methods, and analyzing the impact of these perturbations on the model's output. Moreover, since existing evaluation metrics can have some limitations in measuring the validity of visual explanations for MDE, we additionally introduce the Attribution Fidelity. This metric evaluates the reliability of the feature attribution by assessing their consistency with the predicted depth map. Experimental results demonstrate that Saliency Maps and Integrated Gradients have good performance in highlighting the most important input features for MDE lightweight and deep models, respectively. Furthermore, we show that Attribution Fidelity effectively identifies whether an explainability method fails to produce reliable visual maps, even in scenarios where conventional metrics might suggest satisfactory results.