Why does anxiety cause tightness/pressure in the chest/lower ribs?

Mind-Body connection. Anxiety can cause all kinds of physical symptoms because your mind, emotions, and body are not separate entities. They are richly interwoven and actually a single unit. Your autonomic system, which responds intimately and immediately to what's going on with you at all levels, reaches out and innervates your entire physical being. Your system tried to maintain balance as best it can.