Now, it is a basic theorem of inviscid aerodynamics that a vortex can neither begin nor end in a fluid. It must be either a closed ring, or its ends must trail off to infinity. In the horseshoe vortex description of an airplane, the bound vortex spills off the ends of the wings and becomes a pair of infinitely long trailing wingtip vortices. These are free vortices, with the same strength per unit length as the bound vortex from which they depend.
The effect of a vortex on the air surrounding it is exactly what would be expected. The arrows in figure 1 show the "direction" of the vortex; they induce motion in the surrounding fluid according to the right-hand rule. Thus, the bound vortex running from the left wingtip to the right causes the air in front of the wing to lift up, and produces a downwash behind the wing. The trailing vortices cause the fluid between them to flow downwards, adding to the downwash behind the wing, and they produce an upwash in the regions behind the wing and to the outside of the wing.
When the two aircraft are very far apart, the airflow over them behaves as if they each were alone in the sky. When they are in the configuration shown, however, the trailing aircraft (plane 2 in the figure) is flying such that its left wing is in the upwash induced by the right-hand trailing wingtip vortex form the lead plane. This increases the lift on that part of the wing, at no cost to the lead plane (one can think of it as the trailing plane sucking energy out of the airflow that the lead plane put in; the lead has already thrown the energy away).
This does not completely explain the reduction in drag. Any student of aerodynamics has learned that drag is the sum of friction drag (sometimes called form drag) and induced drag. We could simply say that the induced drag is the result of creating downwash; since in formation the trailing aircraft is creating less downwash, it thereby induces less drag. This is true, but not very satisfying. It may not be intuitive for many people, as well. After all, the trailing wing is still pushing down just as hard on the air moving past - why should the fact that there is also an upwash reduce the drag? If the upwash occurred several feet behind the wing, rather than at the wing, it wouldn't reduce the drag, even if the final downward air velocity was zero.
One way to think about it is to notice that the upwash not only increases the local angle of attack, it also rotates the line relative to which the angle of attack is measured. Since lift is defined relative to the incoming velocity vector, this not only increases the lift but rotates the lift vector forward, as seen in Figure 2. When referenced to the original velocity, the new lift vector has a forward component. Since the overall lift and drag are referenced to the original velocity vector, this acts to reduce the overall drag.
The velocity induced by a vortex is proportional to the inverse of the distance from the vortex. Therefore, the upwash on the wing of the trailing plane in figure 1 is strongest on the left wingtip, and is almost entirely gone at the fuselage. If not corrected for, the effect would be an increase in lift, causing the trailing aircraft to rise, a very strong rolling moment to the right, and a small yawing moment to the right. In order to maintain steady flight in the presence of the vortex, the trailing plane must reduce its angle of attack (to reduce the lift to once again equal the weight), and use control surfaces to balance the lift on the left and right wings. A small amount of rudder may be used to counter the induced yawing moment.
Recall that these remarks are based on assumptions of inviscid flow theory. With no viscosity, the vortex never decays and no energy is lost. This would mean that the trailing plane could follow at an infinite distance and still be able to gain the benefits of formation flight. This is of course not completely true. However, inviscid theory is in fact a good approximation for low-speed aerodynamics, and the vortex effects can be had up to many wingspans distance behind the lead craft. This means that the effects of formation flight are very strongly dependent on lateral relative position, slightly less so on vertical relative position, and vary only slowly with longitudinal position.
Because of these effects and the need to control them, flying in the presence of a trailing vortex is an unstable flight condition. Also, because the effect of the vortex falls off quickly, it is necessary to fly very close to the optimal lateral and vertical position in order to get the maximum benefit. This is complicated by the fact that the vortex from the lead plane is not fixed in the sky; it is moved and deformed by the motion of the ambient air, including the motion induced by the presence of the trailing aircraft itself.