Hi Mike-n-Cheryl,
The pump has a pressure sensor built in. When faucets are opened, the pressure in the line drops and the pump senses the drop and starts pumping. When you close the faucets, the pump continues for a short period until pressure in the line builds back up.
When it cycles on and off with faucets closed, that usually indicates a leak in the plumbing. It's also possible that it can be due to a pump problem, but since you just replaced the pump, it's probably something else.
It's important to locate the leak and fix it as it can lead to extensive damage.