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Part of Covert Affair Season 2, Episode 9:

Auggie: “When were you gonna tell me about Dr. Scott?”

Annie tilted her head slightly and Auggie heard the smile in her voice as she replied, “Sorry, those details are ‘need to know’”. Auggie smiled and nodded slightly. As he listened to Annie’s heels click away, he leaned back on the desk. Her teasing answer did make him laugh and yet … he felt like he did need to know. He needed to know who Dr. Scott was and when Annie had started dating him. The curiosity just wouldn’t leave him alone.

So, what does “need to know” mean here?

In some places (like a military base, or the intelligence community), some information is classified. To gain access this classified information, an individual needs two things:

• They need to have the proper clearance


• They need to have the need to know
  

Let's say that I'm in the Army, and I have a SECRET clearance. I can't walk onto a Navy vessel, and demand to see some classified information, just because I have a SECRET clearance. Yes, I have the proper clearance, but I don't have the need to know.

Similarly, if there is TOP SECRET information in my Army unit, I won't get access to that information, either. I may have the "need to know" (in other words, that information may be directly related to my job), but I don't have the proper clearance.

In the case of the script, I don't think the information is classified – I think the character is essentially saying, "You're being nosy; I don't need to tell you that," but using language that a spy would be very familiar with.

The OED says:

need-to-know, adj. Designating or relating to a principle or policy of telling people only what is deemed necessary for them to know in order to carry out a task effectively.

The phrase seems to have originated in nuclear weapons research in the US:

1951   Science 21 Dec. 656/2   Distribution of scientific reports..is based on a ‘need-to-know’ policy because of security restrictions and the high cost of reproduction and distribution.

1956   Bulletin of the Atomic Scientists XII May 176   What about your definition a little while ago about the need to know? How can you exchange the information, because everybody has to qualify on a need to know?

To tie together the answers from J.R. and Gareth. In the Manhattan project General Groves emphasized to Robert Oppenheimer the need to compartmentalize information such that the team of physicists working on implosion did not "need to know" the progress of the gun-type fission team and vice versa. Oppenheimer argued for and eventually got the ability for the scientists to share ideas freely across teams.
The quote in the question just playfully uses this terminology so as an excuse not to divulge a personal relationship.