What allows burned gasses to leave the cylinders through tubelike chambers into the exhaust pipe?

It depends whether you are talking about a 2-stroke or 4-stroke engine.

A 2-stroke is typically rotary valved where a port opens near bottom dead center of the piston stroke. This port is often in the cylinder wall and this is where the exhaust exits and where the new air/fuel mixture enters.

A 4-Stroke has a camshaft that operates intake and exhaust valves. after the combustion stroke of the piston the cam actuates the exhaust valve. As the valve is open the piston comes up pushing the exhaust gas into the exhaust manifold, sometimes called a header or header pipe.