A current is a flow of charge, so something must push the charge round. A cell or battery is a source of potential difference. It pushes charge round the circuit, and the bigger the potential difference, the bigger the current in the same circuit.
The circuit must also be closed, which means complete, with no gaps. The charge needs a continuous loop of conductor to travel round, from one terminal of the cell, through the components, and back to the other terminal. If the loop is broken, the current is zero.
An open switch makes a gap in the circuit, so no current flows and a lamp stays off. Closing the switch completes the loop, and the current flows at once. The cell does not make the electrons, because they are already in the wires. It only pushes them round.
