Billy Ray’s “Breach” is a fascination because it’s not about a breach at all; it’s about deciphering the makings of a monster.

Far too many spy films have tried to explain everything in excruciating detail — notably the recent and disastrous “The Good Shepherd,” which tackled not only the birth of the CIA, but the catastrophic Bay of Pigs invasion, a multidecade family drama, and the mysteries shrouding Yale’s secret societies. It tried to be all things to all people, and managed in the process to disappoint almost everyone on all accounts. Thankfully, Mr. Ray approaches “Breach” in a fundamentally different manner, denying us the dramatics until we can first appreciate the players who will act out the drama.

Enter Chris Cooper, an actor who has turned being shrewd into an art. When he zeroes in and squints at someone, his wheels churning as he measures credibility, there is an impenetrability that makes him an endlessly fascinating actor to watch. He is inscrutable and detached, and without question the perfect man to play the part of Robert Hanssen, the notorious FBI agent arrested in 2001 and charged with selling American secrets to Moscow for $1.4 million in cash and diamonds over a 15-year period — the same Hanssen who was put in charge of the very investigation launched by the FBI to find him.

Long before we meet Hanssen in “Breach,” we hear about him through the FBI’s Kate Burroughs (Laura Linney) as she removes young hotshot Eric O’Neill (Ryan Phillippe) from his detail and assigns him to monitor the suspected spy (Mr. Cooper). Hanssen is a sexual deviant, she says, and a potential embarrassment to the bureau, and O’Neill needs to set up shop as his assistant, watch his every move, and turn in typed reports every evening on what transpired — every action, every conversation.

Early on, O’Neill assesses that Hanssen might be many things, but sexual deviant doesn’t seem to fit the bill. From his perspective, the man has committed himself to his government, agreeing to a new assignment mere months before retirement because he believes upgrading the security of the bureau’s data information is an essential task. Not only that, Hanssen has raised a happy and loving family, regularly attends Catholic mass, and freely accepts that his inability and unwillingness to play the game of interoffice politics has made for a stagnant career.

He’s not a loner, a godless communist, or an agent with a grudge — O’Neill thinks he’s just a hardworking, misunderstood guy. As O’Neill grows closer to his boss/target and files additional reports, he becomes increasingly skeptical of Burroughs’s judgment, and therefore, his assignment. With the stage set and the characters sufficiently introduced if not deciphered, Burroughs pulls back the curtain one night and reveals everything — from the back rooms filled with dozens of agents pouring over every aspect of Hanssen’s life to the intercepted communications between Hanssen and the Russians and the details behind her larger plan to put Hanssen on a fake assignment and watch him 24/7. If he makes it to retirement without making another information exchange, she explains, the FBI will not have enough evidence to convict him of treason. Naturally O’Neill is taken aback, and remains skeptical of where the truth lies.

Together, Messrs. Cooper and Phillippe deliver performances that carry the movie through its many twists and turns, elevating the drama in “Breach” beyond the mechanics of the spy game to a far more sophisticated study of character and deception. So often, it seems that the movie’s real story is not playing on the surface — where O’Neill is assistant to Hanssen — but underneath, as a seemingly innocent question from O’Neill means far more than Hanssen realizes, and Hanssen’s response seems equal parts honesty, skepticism, and deception.

We, just like O’Neill, understand who and what Hanssen is, but have difficulty in matching the term “traitor” with the man we see standing in front of us. Because of that, we’re fascinated by him in much the same way we are drawn to such screen enigmas as the opera-listening serial killer Dr. Hannibal Lecter or Jack Nicholson’s Colonel Nathan Jessep in “A Few Good Men,” whose poise as an Marine colonel is offset by his belligerent vengefulness. Hanssen, in much the same way, is a contradiction — a man of family and country who, through not only his spying but the bizarre sexual habits that O’Neill finally discovers, betrays both. And it’s a fascinating double life to decipher.

Appropriately enough, the true climax of the film has nothing to do with an act of spying — an event which, as filmed by Mr. Ray (who deftly handled another true-to-life story of deception, 2003’s “Shattered Glass,” about the serial fabricator Stephen Glass), almost seems like an afterthought — but instead is based in a heated scene of drunken arguing, with O’Neill risking his life in one last push to see if Hanssen is really the monster the FBI claims he is.

Mr. Cooper keeps us glued to his every facial expression and makes “Breach” a crackling spy thriller with a human center. When O’Neill later sees Hanssen in cuffs and the latter looks back and asks his young “friend” to pray for him, “Breach” hits a chord that is far more interesting and terrifying than we expect.

It’s easy to cast men like Hanssen as spies and be done with it, as evildoers who walk among us. But it’s far more difficult to engage that rhetoric once you’ve actually met the man. O’Neill, as seen here, finds it difficult to apply such plain labels to this religious, familybound, FBI career man. He struggles to negotiate the Hanssen of Burroughs’s files with the Hanssen he knows from Sunday dinner. And to that end, it’s difficult for us, the citizens of the nation he sold out, to grapple with the prospect that, superficially, this was just your average Joe.

Even at the end — despite the light that’s been shined on what this man did — we sense that for O’Neill and all those around him, Robert Hanssen will forever remain in the shadows.