How do the vortex created by wing tips affects the lift?
The lift over a wing comes from an imbalance of pressure above and below the wing. The higher the pressure, the more force is imparted to the wing. Thus, by generating a higher pressure on the bottom of the wing then on the top, a net upward force is created which offsets the pull of gravity on the airplane. The greater the pressure differential, the more lift is created.

At the end of the wing, there's nothing to prevent the higher pressure underneath from causing air to "spill over" the end of the wing into the lower pressure area. Thus, the pressure at the end of the wing tends to equalize itself, with a corresponding decrease in lift.