To interfere is to try to stop something from happening. If you get into a fight on the street, people passing by might try to interfere so no one gets hurt.
Many countries resent it when other countries interfere (get involved) in their internal affairs, just as people often dislike it when others interfere in their personal affairs. In this sense interfere is the opposite of minding one's business. On airplanes you're not allowed to use cellphones because, it's said, they interfere with the plane’s navigational systems.