I broke my leg sparring two years ago and it has been a rough recovery. One of the things I lost during this time was flexibility, from a full split, and full american splits both ways to something less.
The impact on my kicking and leg control was very noticeable.
With good flexibility the kicks are of course higher but the best benefit is in leg and muscle control I have found.
Recently as my leg and tendons have gotten better, I am again on my way to getting all of my flexibility back, the American splits have returned and the side split (Chinese Split) is coming right along.
I followed the same pattern I did when I was a teenager, streatch before a work out to prevent injury (Bernard is right dont do it cold, just warm the muscles up by running in place or something) then after the workout when the muscles are really loose and warm streatch for gain. You will find if you follow this methodology, most people show great results. The best progress is made right after a good workout.