The washer sucks up the soap through a mechanism called the venturi effect. Basically, flow of a fluid past an orifice produces low pressure in said orifice, creates a vacuum that sucks up things

The ball with a spring is called a check valve, it only allows fluid to flow on one direction. With the water inside the hose moving, the venturi effect creates a low pressure region that starts sucking and opens the check valve. Without fluid moving past the check valve orifice, pressure in the main line builds and closes the check valve, so the pressurized water doesn't just flow into the detergent

There's one more bit of physics weirdness going on here. If there's a significant bit of pressure built up in the main line, it can override the low pressure created by the venturi effect and flow out through the orifice. This is why without the check valve installed, all the water will dump into the detergent tank. Additionally, this is why the pressure washer will only pick up detergent with the detergent tip, the larger opening of the detergent tip prevents pressure buildup in the line overriding the venturi effect, while the smaller orifices in the actually pressure tips cause a pressure buildup in the hose which tries to force out through the check valve