Neuroscience is complex; this is a simplification. To summarize: yes, breathing techniques and other meditation and biofeedback methods have been shown to be effective for reducing anxiety and panic, as has been cognitive behavioral therapy. Psychotherapy is a critical part of treatment of anxiety disorders. Some people benefit from medications like SSRIs as well. BUT because anxiety might be a symptom of a medical condition, it's important to see a doctor to be formally diagnosed, and receive individualized treatment recommendations.
Generalized anxiety disorder is, at its root, over-activity of the sympathetic nervous system. It is when the normal human "fight or flight" response is extended and/or exaggerated. Panic attacks occur when that gets stuck in a positive feedback loop.
Physical symptoms of anxiety or panic are due to this sympathetic hyper-activation: increased heart rate, blood pressure, respirations, alertness, sensory awareness (including pain), sweating, speed of reaction, gastrointestinal upset, etc. It helps to understand that anxiety and panic disorders develop more frequently in people who have experienced severe traumatic stressors, especially during childhood, during which their neurological system becomes conditioned to overreact. During "fight or flight" much of our mental processing gets caught up in the limbic system (center of emotion) rather than engaging the frontal cortex (center of logical reasoning), which is why anxiety can impair clarity of thought, decision-making, and memory.
Deep breathing, mediation, and certain biofeedback techniques can activate the parasympathetic nervous system, thereby decreasing sympathetic tone. With practice it can interrupt even the positive feedback loop of panic. This not only reduces the physical symptoms, it allows our frontal cortex to engage in processing input and thoughts more logically. The long term impact of these techniques can be profound.
Resources below are as a supplement for professional evaluation and treatment.