Could you please explain how this works to a person of execrable scientific understanding?
That's a wonderful word, execrable. I don't think it describes your scientific understanding. In many ways you have a scientific heart, Faith.
I don't know the specific technique used, I'll see if I can find some primary literature, but it works because the surname is traditionally passed down the paternal line, so people that share a surname are likely to be related (though with common names, it doesn't work).
As surnames are passed paternally, so is the Y chromosome. The idea is that they look at the chromosome and compare it to others they have on record. The ones that share the highest levels of similarity are the most related, and are most likely to share the same surnames.
It's not perfect for many reasons, but it can help focus on certain suspects, asking them to submit to DNA testing.
I didn't explain that very well I know, but do you get the gist?