How does a helicopter move forward?

When the stick is moved by the pilot, it moves an assembly called a swashplate accordingly. The swashplate is at the rotor base. When this moves, it changes the pitch of the rotor blades. As the pitch changes, the orientation of lift produced changes as well. Thus, when the stick is moved forward, the blade pitch is affected by the swashplate to re-orient the lift forward, instead of directly vertical, and the helicoptor is pulled in a forward direction.