As I understand it, it's a holdover from NextStep (which OS X is based on), and NextStep did it to support NetBooting. The idea was that you could boot from a network-hosted volume (probably read-only, and certainly shared with other computers), and early in the boot process mount a local (writable) volume on /private; as g mentioned, this allowed runtime-modification of /var and /tmp, as well as per-computer settings in /etc.
This isn't needed anymore, as Apple's current NetBoot system uses a shadow disk image to store changes anywhere on the boot volume. But some programs/docs/etc now assume the files live under /private, so it'd be too much trouble to switch them back...
Update: since I wrote this, Apple has stopped supporting NetBoot, so the original purpose of /private is even more obsolete. However, in macOS Catalina (version 10.15), they've added a new volume split. In this case it's for security rather than to support NetBoot, but it works in a fairly similar way.
Catalina's system volume is mounted read-only, with a read-write volume mounted at /System/Library/Data (analogous to the old system that mounted a RW volume at /private), and "firmlinks" making parts of the RW volume appear at their usual locations in the filesystem (again, analogous to the symbolic links that make parts of /private appear at their usual locations). For example, /Users is now a firmlink to /System/Library/Data/Users. The Eclectic Light Company has a good summary.
Catalina also still has the symbolic links to /private; thus, when you access /etc on Catalina, it follows the symlink to /private/etc, and then the firmlink to /System/Library/Data/private/etc
After doing some research, I ended up writing a shell script and using launchd to have it execute at boot time.
The script I use (/usr/local/bin/empty-tmp.sh) :
#!/bin/bash
# Delete files (other than directories) that haven't been modified in more than 1 day
find /Volumes/MacintosHD/private/tmp -not -type d -ctime +1d -delete
# Delete empty directores that have been created more than 1 day ago
find /Volumes/MacintoshHD/private/tmp -d -type d -empty -mindepth 1 -Btime +1d -delete
and the property list (in /Library/LaunchDeamons) :
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
<dict>
<key>Label</key>
<string>com.samric.emptytmp</string>
<key>ProgramArguments</key>
<array>
<string>/bin/bash</string>
<string>-c</string>
<string>/bin/wait4path /Volumes/MacintoshHD/private/tmp/ && /usr/local/bin/empty-tmp.sh</string>
</array>
<key>RunAtLoad</key>
<true/>
<key>LaunchOnlyOnce</key>
<true/>
</dict>
</plist>
Now the key is using wait4path to make sure the secondary HD (MacintoshHD) is properly mounted before invoking the script. Failing to do so will result is the script not deleting any files since they won't be accessible.
Best Answer
Here is how my Catalina machine looks:
So:
/tmpis a softlink to/private/tmp/private/tmpis a directory.