Сейчас is like English "right now", something is happening at this moment in time. Теперь is used more to contrast with another point in time, things were one way before, but they're different now. In this sentence, rather than meaning you need to do this action right now, the English suggests that something, probably bad, has happened, which now needs to be explained to Mom, as opposed to before when it hadn't happened, so теперь is used. There's better explanations than that around, if you search discussion.
Or it was already explained to one parent, who is now making the kid explain it to the other parent to rub it in further how badly they messed up. ;-)
Теперь is a combination of этот /that/ + первый /first/ and has a function similar to 'hereafter' in English. In contrast, as already mentioned by a previous comment, сейчас (сей /here, now/ + час /time, hour/) means exactly 'at present', 'right now' and bears more narrower sense.
What about necessary to use the possessive pronoun "your" before the word mother. And now explain it to your mother, As usual I keep a pen in my hand. He took the umbrella his right hand.
You can use the possessive pronoun too, but depending on the context, there would be no need.
"And explain it to Mom now" seems ok to me, i think it should be accepted. Although it is more common to keep "and now" together, just like "а теперь".
его is он's genitive. Это's would be этого. You would use it if implying that there's an animate thing to explain. But in the phrase, there's probaly an inanimate object being explained.