replaced battery last week and now the car is not starting and there is a clicking sound when I try to start it

Hi, I had my car serviced about a week and a half ago and i came to start the car last week and replaced the battery and it did not start. It made a fast clicking sound. I called the place where I had my car serviced and they want more money to fix something they most likely did because my car has never done this before. So i went to YouTube, apparently rust or something on the battery causes a car not to start, seeing as my battery is new, it did not have this problem, but it had a yellow substance on it so i cleaned it with bicarb soda and water. However, my car still did not start. The battery connectors are not loose, everything is intact. Please help ASAP. I need my car!