Anlehnung means contact, or connection, and it refers to contact between the horse's mouth and the rider's hands via the bit and reins.  

So the translation of the word is simple. But the application is the most difficult yet. You see, a contact cannot just be imposed by the rider taking up the reins, it has to be a joint effort. It’s like when you meet somebody and you want to shake hands. You hold out your hand as an invitation. The other person gets to decide if they wish to accept that invitation. If they do, they reach out to you, and you have anlehnung. If not, you don’t pull their hand out of their pocket and force them to shake yours!  
So the challenge to this step is in getting the horse to understand that he is to reach out to the rider, and that he wants to reach out to the rider! Clearly, in order for the horse to do this, he must be relaxed and trust his rider, so losgelassenheit must be in place.  
In classical training horses are started on a longe line to develop their muscles and teach them some basic lessons about balance, voice commands, obedience, and more. Side-reins are used while longeing so that the horse can begin to learn about anlehnung prior to carrying a rider. Side-reins have developed a bad reputation due to misuse, but properly used they are a truly valuable tool.   
Side-reins should be attached to the saddle or longeing surcingle at a height that will allow the side-reins to be horizontal when the horse carries his head at its natural height, and they should be long, but not so long that the horse can’t reach them. The horse will soon realize that the side-reins provide some gentle support at times when his balance falters a little, and he’ll begin to seek their reassuring contact, like a security blanket. At this point, the side-reins may need to be shortened a bit so that the horse can find them more easily. By the time the rider gets on, seeking contact with the bit should be second nature.  
In order for the horse to develop anlehnung under saddle, his rider must have considerable skill. The rider must sit lightly and in balance, allowing the horse to be relaxed and losgelassen, particularly through the back muscles. If the back is tight and tense, there can be no anlehnung. The rider must also have excellent hands. The horse’s mouth is extremely sensitive. In order for the horse to reach into a contact with the bit, he must be able to trust his rider to be kind and considerate.  
Unfortunately many riders have more ambition than skill or patience and they do impose a contact on their horses. At best, these people are guilty of “hand-riding” and at worst, outright abuse. For more on problems with contact, go to this website.  Entitled Sustainable Dressage, it is actually about equine biomechanics which are the same for all horses regardless of discipline.
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One way to check whether the horse is stretching into contact correctly is to loosen the reins.  If he is truly reaching for your hands, he will follow the reins down into a long and low posture.  If the contact is correct it feels as though the reins are sticks, pushing the horse's head forward.  

If the horse is held into a frame by the riders hands, then when the reins lengthen he may go above the bit, behind the bit, or some other variation.  Again, go here for more information.