Providing timely and actionable feedback is crucial for effective collaboration, learning, and coordination within teams. However, many teams face challenges in receiving feedback that aligns with their goals and promotes cohesion. We introduce tAIfa (``Team AI Feedback Assistant''), an AI agent that uses Large Language Models (LLMs) to provide personalized, automated feedback to teams and their members. tAIfa analyzes team interactions, identifies strengths and areas for improvement, and delivers targeted feedback based on communication patterns. We conducted a between-subjects study with 18 teams testing whether using tAIfa impacted their teamwork. Our findings show that tAIfa improved communication and contributions within the teams. This paper contributes to the Human-AI Interaction literature by presenting a computational framework that integrates LLMs to provide automated feedback, introducing tAIfa as a tool to enhance team engagement and cohesion, and providing insights into future AI applications to support team collaboration.