Allowing your body to move naturally instead of trying to control every muscle in each movement produces a very fluid and graceful dancing action. Men especially get caught up in trying to identify and move with very specific muscles to produce particular movements. This produces a stiff and robotic action, look and feeling. The ballroom dance frame is allowed to conform to each partner, not maintain a ridged partnership relationship. Ultimately we want the partners to be allowed to mold to each other in the partnership and in the individual movements.
Men, in general, learn mechanically so they try to identify what muscles to move in each movement. This mental involvement is what causes the movement to be rather robotic. On the other hand women generally learn by feeling. They don't try to figure out what muscles to move. They can reproduce the movement by how it feels. When I teach women I usually dance them through the movement to get them to feel it, then I tell them how it is done. Men need to practice the movement until it becomes muscle memory and can be executed with out thinking. Eventually it smoothes out and becomes natural.
Most often you cannot determine what causes a particular movement by watching someone dance it. You may think you know, but when you try to dance it yourself, the result is not the same. It is important that you know exactly how it works, whether you learn my feeling or mechanically, to get the best results. This is one reason why "group dance lessons" don't get the expected results.
Exactly what does "Allow the body to move." mean? The "body" in this instance, may be all or specific parts of the body (arms, legs, feet, head, etc). Generally speaking, it means to allow other forces besides your muscle power to participate in the movement. Some of these forces are: gravity, centrifugal force, momentum, counter balance, the pendulum action, continuation of movement, your partner, etc.
An example in Silver Waltz would be what I call "Power and Glide". The muscles produce the movement (Power) on the "1" (heel step) and the momentum created is allowed to continue (Glide) through steps "2" and "3" (ball steps). This use of power creates a smooth and controllable movement. Powering on each step produces a more mechanical and robotic action. Silver Waltz also has something called "Leg Swing". This is created by allowing the leg to swing from one side of the body (in the direction of movement) to the other side of the body during "rise" on the 2nd step. If you allow your leg to swing naturally, gravity will place it in precisely the correct spot to maintain proper balance of the dancer and the partnership. You don't have to guide the leg with muscles (gravity does it for you).
Let's talk a little bit about "Latin/Cuban" motion. Latin/Cuban motion is composed primarily of rotations in the legs. The leg in Latin/Rhythm dancing is defined as: "from the bottom of the foot to the bottom of the rib cage". The Latin/Rhythm leg includes the foot, the leg, the hip and the soft area from the top of the hip to the bottom of the rib cage.
The hip action is a result of rotation in the supporting leg. The weight transfer of the body from one foot to the next foot, is accomplished through the ankles. The foot work on all steps is "ball" - "flat". As the new foot receives the body weight ("ball - flat"), allow the hips to rotate naturally (settle) creating the Latin/Cuban motion. I'm sure you have heard the expression of being "knock kneed" and the feet move with "inside edges" (actually "inside balls") in Latin/Cuban movement. Allow the "inside edge" and "knock kneed" action to occur as a result of the shape of the hips.
Here are a few more examples:
The concept of "allowing" the body to move creates a softer, more natural and fluid movement of the partnership. The movement is less precise from a muscle stand point because more natural forces are allowed to be involved. This allows the two partners to evolve into a single entity, the infamous "two-headed four-legged animal". The look and feel of your dancing will be changed forever.