The obvious reason to move the weights slowly, is so that you don't either drop the weight on yourself or on someone else. Swinging a weight around has obvious dangers to it, but there are others that we want to look at here.
In a lot of sports like soccer, football and hockey, there is a considerable about of speed involved, and it is often combined with weight. This creates a considerable amount of force (Force = Mass (weight) X Acceleration (Speed)). And force when it it applied to a joint like the one in our knees, can cause injury. Muscles can build up more force than the joints can handle when movement is done using speed. In the context of lifting weights, lifting weights fast, can increase the amount of force, and sometimes impose that force on the joints that cannot support it. This is where speed can cause injury to a joint in weight lifting. There are certainly other ways you can cause damage to the tissues in your body lifting weights, and not always, moving a weight quickly will automatically cause injury, but we are trying to do exercise in the Simply Stronger program in a way that is the safest possible, without compromising its efficacy.
An other reason going slower is safer, is that if you were to do a movement in the program that could cause you injury or pain, the faster you were moving the less time you'd have to react to prevent the injury. As an example, if I were to stretch my index finger back so that I was trying to touch the tip of my finger to the back of my wrist (something I promise I cannot do!) I would get to a point - maybe half way there - that I would start to feel a stretch in the palm of my hand. If I kept going, this stretch feeling would turn into a pain feeling (my brain starting to warn me "um... you might not want to keep doing that...") and if I kept going any further, I'd probably stop before I tore something in my finger. That's IF I did it slowly. If I moved slowly I would be able to stop before I got to that point that was too far. But if I did it in one fast movement (something I don't recommend) then I might go past that point of "warning-pain" right into injury. The slower I move, the more reaction time I have to feel and say to myself "um...you might not want to keep doing that...". So if you were doing a movement in the program that could cause you injury or pain, the faster you were moving the less time you'd have to react to prevent the injury. So, if you were to do a movement in the program that could cause you injury or pain, the faster you were moving the less time you'd have to react to prevent the injury.
These are just a couple of reasons it's a good idea to slow down the movement when you are lifting weights.