Many people believe that a run-up helps them to jump higher.
Is this true?
(While it seems natural to do this, I'm having trouble believing that horizontal speed can help one jump vertically.)
It is true.
See Young, Warren, G. Wilson, and C. Byrne. "Relationship between strength qualities and performance in standing and run-up vertical jumps." The Journal of sports medicine and physical fitness 39 (2000): 285-93.
Subjects were 29 males aged 19 to 34 with at least one year experience in a sport requiring sprinting or jumping.
The reason why is reactive strength. (Ball, Nick B., and Sara Zanetti. "Relationship between reactive strength variables in horizontal and vertical drop jumps." The Journal of Strength & Conditioning Research 26.5 (2012): 1407-1412.)
Reactive strength is an athlete's ability to use the "stretch-shortening cycle (SSC) to increase subsequent force production".
A drop jump is a common way to display and assess reactive strength.
Drop jumping is a plyometric activity that involves stepping from a predetermined height, landing, and immediately performing a maximum jump.
A runup similarly allows a jumper to make use of the stretch shortening cycle to increase subsequent force production.
From (Young et al. 2000):
reactive strength is relatively more important for jumping from a run-up than for the standing VJ
Also, not everyone is able to make effective use of the stretch shortening cycle — some peoples' drop jumps and runup jumps are similar in height to their standing jumps.