The rising sophistication of Cybersecurity threats demands advanced Intrusion Detection Systems (IDS) capable of identifying abnormal network traffic with both accuracy and speed. This Research proposed FNR-IDS, a hybrid detection model that integrates Fuzzy Logic, Deep Neural Networks (DNNs), and RSA encryption to enhance both intrusion detection and data confidentiality. Using the UNSW-NB15 dataset, we apply Random Forest-based feature selection to extract key traffic attributes. A fuzzy inference engine evaluates the threat level and triggers RSA encryption for high-risk instances, while the DNN classifier ensures accurate detection. The integration of interpretability, learning capability, and real-time encryption addresses critical gaps in conventional IDS models. Experimental results show that FNR-IDS achieves 87% accuracy with an F1-score of 90% at the optimal threshold, confirming its effectiveness in detecting and mitigating modern cyberattacks. The proposed model of this article offers a robust, explainable, and secure framework for next-generation intrusion detection.