Question: > 
> I told you never to go nowhere without me!
> 
> 
> 

According to Grammarly, the above sentence contains a double negative and should be modified to something like, "I told you never to go anywhere without me!"

Doesn't the original sentence mean the person addressed as "you" in the sentence must stay here?

Answer: 
The sentence contains a double-negative, (or perhaps actually a triple negative) and this means it is unclear.

It would be taken to mean "I told you never to go *anywhere* without me." This does mean "stay here". This is because the "logical" removal of the double negative (don't go nowhere -> do go somewhere) doesn't work here.

If you try to apply "logic" you reach something like "Always go somewhere without me." Which is clearly nonsense.

So you would assume a dialect in which the double negative is used and the meaning is "I told you to stay here until I come back."

Learners should **avoid** this kind of double-negative, because it is unclear and considered to be a mistake in careful English.
In lots of non-standard English dialects the double negative is fine.

Not in standard English, though: in standard English, Grammarly is correct.

Double negatives are possible in standard English, but they often add up to an affirmative. They're sometimes useful in conversation, often to contradict negative assertions by one's conversation partner.

Grumpy teacher to student: "You're going nowhere, Jimmy."
Student: "I'm *not* going *nowhere*; I'm just going nowhere you'd like me to go."

They can also be seen as two single negatives, each in a separate clause:
"She did*n't* say you could*n't* go; she only said you couldn't bring your laptop with you."

I'm not sure I understand the last sentence in your question.