What does having your pants on fire have to do with being a liar, other than the fact that it rhymes?
What do cats have to do with fiddles, other than sounding silly and amusing small children?
I haven’t exactly done research into the meanings of children’s rhymes, but I don’t think there’s any “deep meaning” to “liar, liar, pants on fire.” It just sounds good. It’s a nonsense verse.
Maybe because sometimes, when people lie, the fidget or move around a lot? Kind of like someone who has their pants on fire, I would think. Although, if my pants were on fire, I don’t think I’d lie about it 
As for the next line, at least the way I knew it, “hanging from a telephone wire” probably just means its obvious. In other words, “you’re lieing and I can tell because you’re moving around and fidgetting and it’s just so obvious to anyone looking”.