A semicolon is best used to link two phrases that grammatically could be separate sentences, but that need to be linked together somehow to show their proper meaning. For example:

I don't care what the dictionary says; I like the word "irregardless."

If you were to replace the semicolon with a period, it would give the impression that there are two thoughts here, each of which could stand on its own. That would give the impression that in general I don't care what the dictionary says. Actually, that's not true: In general, I do care what the dictionary says, but this case is an exception. To get the proper meaning for the first part, you have to interpret it in light of the second part.

In my experience, grammar books often give poor examples of the principles they are trying to teach.