As a typical entity of MEC (Mobile Edge Computing), 5G CPE (Customer Premise Equipment)/HGU (Home Gateway Unit) has proven to be a promising alternative to traditional Smart Home Gateway. Network TC (Traffic Classification) is a vital service quality assurance and security management method for communication networks, which has become a crucial functional entity in 5G CPE/HGU. In recent years, many researchers have applied Machine Learning or Deep Learning (DL) to TC, namely AI-TC, to improve its performance. However, AI-TC faces challenges, including data dependency, resource-intensive traffic labeling, and user privacy concerns. The limited computing resources of 5G CPE further complicate efficient classification. Moreover, the "black box" nature of AI-TC models raises transparency and credibility issues. The paper proposes the FedEdge AI-TC framework, leveraging Federated Learning (FL) for reliable Network TC in 5G CPE. FL ensures privacy by employing local training, model parameter iteration, and centralized training. A semi-supervised TC algorithm based on Variational Auto-Encoder (VAE) and convolutional neural network (CNN) reduces data dependency while maintaining accuracy. To optimize model light-weight deployment, the paper introduces XAI-Pruning, an AI model compression method combined with DL model interpretability. Experimental evaluation demonstrates FedEdge AI-TC's superiority over benchmarks in terms of accuracy and efficient TC performance. The framework enhances user privacy and model credibility, offering a comprehensive solution for dependable and transparent Network TC in 5G CPE, thus enhancing service quality and security.