Basically you can picture layer like storeys in a house.
If you put your traces on different layers, they can cross each others path without making a connection.
If you want a connection between traces (or planes) on different layers you need a plated hole in the pcb usually a via or component hole.
That way you can pack your board very tightly with components and traces. It is hoewever considered good practice to keep your layercount as low as possible.
2 layers is pretty much standard these days for hobbyprojects and the like.
Everything above 4 can get expensive rather quickly..