The /run directory is the companion directory to /var/run. Like for example /bin is the companion of /usr/bin.
That means that daemons like systemd and udev, which are started very early in the boot process - and perhaps before /var/run is available (i.e. mounted) - have with /run a standardized file system location available where they can store runtime information.
Like /bin contains important programs, which may be needed in the boot process before /usr is available (in case it is on its own filesystem).
The /run idea is a relatively new idea/standard, one supporter is the developer of systemd.