How do I use this file?

Using the my custom ships shouldn't be too incredibly hard.

You should be able to get this to work by doing the following:

Make sure you have a copy of shipedit.exe
Download the custom ship file
Find your Tyrian directory on your hard drive using something like My Computer or Windows Explorer
Find the file called "user.shp" and rename it to something else that you will remember (all you're doing here is backing up your current ship file)
Copy the ZIP file into this directory and extract "user.shp" into the directory
Run Shipedit.exe in your Tyrian directory and exit the program - this will compile newsh$.shp for Tyrian (I would give you my copy, but it's version specific and shipedit compiles it for you anyway)
When you're in the game use Tab+{ship #} to use the custom ships.