Nothing's wrong with your Mac. OpenOffice.Org runs in the X11 windowing environment on Macs. It always has. The NeoOffice variant runs inside Aqua. But OOo does not and never has. This has nothing to do with Tiger.
Two choices:
1) Get Neo instead of OOo
2) Install X11 as OOo suggested you should. To do this, insert the OS X install DVD in the drive (disc 1), open it in Finder, scroll down to the bottom of the main window, and run the optional installs package. When you get to the part that asks you what you want to install, make sure X11 is on the to do list. Do not accidentally install the X11 developer tools, as this is something else entirely. If you've succeeded, you will have an application called X11.app in your utilities (I think) folder. And OOo will run correctly.