When you run the water in the kitchen and bathroom and flush the toilet the water comes out through the pipe. Why?

What pipe? Drainage water flows through a lot of pipes.

What pipe? Drainage water flows through a lot of pipes.

It's coming out of the drainage pipe between the washer and dryer...

You have a clog or partial one in the drain line. It is after the washer but since we don't know your layout hard to tell you where to start. Look for a cleanout in the line and start there. Washers can dump a lot of lint into a drain line.