While the primary goal of strength training is to put tension on the musculature to elicit a desired adaptation the goal of exercise in general is to improve your quality of life not hinder it. If you get injured while exercising it is not improving your quality of life it's hindering it and therefore it's not really exercise. By moving slow in your resistance training workouts you are minimizing any potential for your muscle to not be under tension which also means you are eliminating any potential for your joints to be under tension. The large majority of resistance training injuries happen because the joint gives out rather than the muscle. You muscle can handle a lot of tension, that is their job, but your joints cannot handle a lot of tension. By doing fast "explosive" exercises you are putting your joints and ligaments under a lot of stress and putting yourself at a greater risk of injury during your exercise or outside of your exercise.
In order to really understand the injury risk associated with explosive style weight lifting it's important to look at some of the research on this topic. A meta-analysis by Southampton Solent University titled "Explosive Exercise in Sports Training, A Critical Review" the risks of explosive weight training were outlined. The authors state "injuries to the wrist, elbow and shoulder were commonplace when individuals performed fast, Olympic-style lifting." In addition it was observed that from their 3,132 subject cohort there were 390 cases (22.7%) of lumbar spondylolysis resulting from fast weightlifting. This was in experienced professional weight lifters meaning they were well versed in the proper technique of the exercises but were still at a very high risk of lumbar injury. Lifting at fast speeds greatly increases the shear forces on the lumbar region and can lead to spondylolysis. Other studies have found that up to 44% of professional Olympic weight lifters had spondylolysis compared to 5% of the normal population. In a study of weight training injuries in football players, Risser et al. found that 60% of their sample who performed Olympic-style lifts suffered from low back problems, compared to only 14.3% of athletes who did not perform such movements. Researchers Konig and Biener noted that 68% of their sample of Olympic lifters had suffered an injury as a result of their weight lifting, and 10% of these required at least 4 weeks’ recovery before being able to return to lifting weights. Granhed and Morelli also found that 46% of retired weight lifters had physical problems caused by their lifting. Bryzcki even cites the case of an experienced athlete who fractured both of his wrists when attempting a power clean. It is clear the risk of doing fast "explosive" movements when weight lifting is dangerous even for the most experienced weight lifters. In order to be able to engage in strength training long term it's vital that you are using slow and controlled speeds during your exercise!