For a reaction to happen, current (electrons) must flow through a complete circuit. Here, you have two separate reactions. One is the production of hydrogen, which requires additional energy to occur (we call this non-spontaneous). The other reaction is the reaction of zinc and copper, which happens without added any energy (we call this a spontaneous reaction). So there are actually two reactions happening at the same time. One is the reaction of zinc and copper, which liberates energy that is used to power the reaction to create hydrogen gas. In the same way, you could put a battery into water and create hydrogen gas. The role of the zinc and copper is to provide electric energy to make the hydrogen gas.