You say :

If someone can explain to me what exactly the security threat of running old software is, I’d appreciate it greatly. I know it’s a threat; I just don’t know why or what sorts of bad things could happen because of the holes in the software.

I’m sure there are people who can explain this better than I, but here goes. The problem is that no software is perfectly secure, and the older the version of the software is, the more likely it is that the imperfections are known. Bad guys that know the imperfections exploit them, while the good guys patch them, resulting in a new release that eliminates that known problem.

Here is an example of a WP 2.3 exploit that your site would apparently be open to. This sounds awfully similar to something that happened to an un-updated Movable Type installation I once had, where mysterious directories crammed full of spammy junk appeared. In short, your site can end up hosting files that are malicious, spammy, or that send people to malicious sites. And it can take a while to notice that.