Traffic prediction represents one of the crucial tasks for smartly optimizing the mobile network. Recently, Artificial Intelligence (AI) has attracted attention to solve this problem thanks to its ability in cognizing the state of the mobile network and make intelligent decisions. Research on this topic has concentrated on making predictions in a centralized fashion, i.e., by collecting data from the different network elements and process them in a cloud center. This translates into inefficiencies due to the large amount of data transmissions and computations required, leading to high energy consumption. In this work, we investigate a fully decentralized AI solution for mobile traffic prediction that allows data to be kept locally, reducing energy consumption through collaboration among the base station sites. To do so, we propose a novel prediction framework based on edge computing and Deep Transfer Learning (DTL) techniques, using datasets obtained at the edge through a large measurement campaign. Two main Deep Learning architectures are designed based on Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) and tested under different training conditions. Simulation results show that the CNN architectures outperform the RNNs in accuracy and consume less energy. In both scenarios, DTL contributes to an accuracy enhancement in 85% of the examined cases compared to their stand-alone counterparts. Additionally, DTL significantly reduces computational complexity and energy consumption during training, resulting in a reduction of the energy footprint by 60% for CNNs and 90% for RNNs. Finally, two cutting-edge eXplainable Artificial Intelligence techniques are employed to interpret the derived learning models.