Question: Do they mean something like "please go! You must leave!" or could it be "We assure you that he left"?

Answer: 
In American English, there is no ambiguity. There is no issue with it being idiomatic; it is. Your sentence is in the indicative for us:

> 
> They insisted that he left.
> 
> 
> 

That version *always* means that they are assuring you that he has indeed departed. It is a done deal.

Had we meant the other thing, we would have said this, using an untensed verb in the subordinate clause to indicate subjunctive effect:

> 
> They insisted that he leave.
> 
> 
> 

Which means that they had demanded his departure. Whether he actually left or not is unstated.
Noting that @tchrist assures us there can be no ambiguity in AmE, I feel I should post the contrary position. Which I can't substantiate as representing BrE in general, but it's certainly how ***I*** see it.

There's no doubt the untensed "infinitive/subjunctive" *They insisted [that] he leave* only ever means that what they demanded (of *him*, or whoever controlled his actions) was that he ***should*** leave.

But to me at least,...

> 
> [*She insisted that we **went** into the house and **shared** the meal they were just about to have.*](https://www.google.com/search?q=%22She%20insisted%20that%20we%20went%20into%22&btnG=Search%20Books&tbm=bks&tbo=1)
> 
> 
> 

...is perfectly acceptable. But if it weren't for this question reminding me of a possible (perverse) interpretation, it would never occur to me to think it might mean she insistently claimed later that this is what happened.

In short, I don't think it's particularly uncommon (or generally perceived to be a significant error) to use *X insisted that Y **Z'ed*** to mean they were adamant that *Y **should** do Z* (was required to, at the time). The alternative interpretation - insistence that *Y **did in fact** do Z* - may be equally credible in some contexts, so I would simply say the construction itself is inherently ambiguous.