Think about the number of places where a flat object meets another flat object within your 3D shapes. For example, the cone is made up of a rectangle rolled up and a circle. The rectangle and circle meet up at one point, the entire way around the circle.
Therefore, the cone has one edge.
A cylinder is made up of one rectangle and two circles. The rectangle is again rolled. It meets with the first circle similarly to the cone situation, so there is one edge. What happens with the other circle? Hopefully this helps!