Many people have heard or even repeat the phrase "speed kills" when it comes to sports. This is true, but often it is being said to demonstrate that speed training is more important to strength training when it comes to sports performance. Speed and strength are certainly linked, but it is absolutely not a zero-sum contest between the two.
Running speed ultimately comes down to two variables, stride length and stride frequency. Increasing one or both will produce an increase in speed. Stride frequency is determined by how fast your muscles can activate and cycle through the running motion while stride frequency is determined by, first limb length, but also how much force can be put into the ground to propel the body forward as far as possible before the next step. These two factors are where muscular speed and strength meet.
Knowing how running works the next question is how do we improve it. First muscle firing speed is largely determined by muscle fiber type distribution, which unfortunately is largely genetic, but can be altered in a limited capacity by consistent high velocity training. Next we can improve the coordination of muscle firing when running. This is where running technique comes into play. Learning proper running mechanics allows the athlete to move at optimal efficiency to move as fast as possible. When muscles fire in a highly coordinated fashion the body is able to maximize the resultant speed. Last comes strength, which is the primary point of this post. Several studies examining how to improve running speed have determined that increases in speed can be improved most by increases in strength. By increasing the force put into the ground when running increases the force with which the athlete can propel their body forward. Strength as a physiological characteristic is also highly trainable so athletes are able to make strength improvements that result in speed improvements as well for most of their athletic career before they start to plateau, while on the other hand maximum muscle firing speed is reached much earlier in an athlete's career.
During the early days of strength training in sports coaches noticed that people with a lot of muscle mass were slow, inflexible, and poorly conditioned. They came to the conclusion that large amounts of muscle mass was the cause. This could not have been further from the truth and this myth has been toxic ever since. The true reason was that most of the people observed to have large amounts of muscle mass at the time were body builders that did not train for speed, flexibility, or conditioning. Fortunately we have seen the competition results and the truth is obvious. Look at the fastest people in the world (100m sprinters), nearly all are built like body builders.
To close out, the belief that improvements in strength create improvements in speed is well supporting in both the research and practical cases of competitive athletes. When done along with proper running mechanics improvements in strength will directly produce improvements in speed. And like the title says strength = speed!