Why do I have such a strong pain all over my gums?

Good hygene. If after a couple weeks of vigorous daily brushing and flossing you still have bleeding, seek an early dental check up.
Most Likely Perio. Without more information i can't be specific, but the most likely answer would be a form of periodontal disease. There is also a stress induced disorder called anug (also known as "trenchmouth " from ww1 soldiers would develop while in the trench's of ground warfare) which can cause gum pain throughout mouth. Best idea is to see a dentist asap for an exam and diagnosis.