What does “Come-to Jesus (moment / stage / meeting)” mean? Is it a popular word?

Come-to-Jesus means, in a general sense, to come to or return to core principles.

It comes from making a commitment or conversion to Christianity.

In both instances, a come-to-Jesus moment may include a public display where one shows that they have accepted their need to change.

When you face the figurative come-to-Jesus moment in your life (at work for example), you face the error in your ways. (You accept the fact that you have erred. A typical expression in common with the literal and figurative expression would be that you have strayed). Once you have recognized your error you then make a commitment to return to core principles.

In your example about the CIA* running amok, it means they have strayed from their defined behavior. A come-to-Jesus moment should bring them back to the fold or change their behavior to be more compliant with what is expected of them.

As to whether its a common or popular phrase, I would say it's well known in the U.S. and used occasionally in private communications. It's one of those phrases where, when you hear it or read it, if you don't already know what it means, you may still have a fairly good sense of what it means without asking. With religion being such a charged topic here, I wouldn't expect to hear it coming from our news networks (written or broadcast) unless they were quoting someone. It's an informal expression, so you'll hear it in informal dialog or see it in informal writing. You may see it within quotation marks in anything formal.

(*The CIA is referred to as "Langley" because their headquarters is in Langley, Virginia.)

It means a reckoning such as when an employee is not meeting their assigned goals, they might be called into their supervisor's office for a "come-to-Jesus" meeting where they will be judged on their actions.

The definition according Jargondatabase.com:

"Come To Jesus Meeting"

"A time when a polite ultimatum is given, generally followed by a less polite ultimatum, then a threat. Drug and alcohol "interventions" are often referred to as "Come to Jesus Meetings"."