Direct connection in a zinc copper battery
A close circuit is needed for electrons to flow. The electrolyte and salt bridge serves to complete the circuit. Additionally, the salt bridge also serves to function of maintain electrical neutrality in each half-cell, preventing the accumulation of charges, aiding the flow of electrons. 

If charges were to accumulate in each half-cell, the potential difference between the two half-cells, which is the key driver of electrons, would decrease over time to 0 V and no electrons would flow. 

There have been many discussions on this site. To cite a few: 
What is the real purpose of the salt bridge in an electrochemical cell? (look at the comments)

Why is it important to use a salt bridge in a voltaic cell? Can a wire be used?

electrochemical (Galvanic) cell solution purpose