The informal phrase to go ballistic means:
– to become wildly or explosively angry;
– to become highly excited or enthusiastic;
– to intensify rapidly and especially alarmingly.

The specific reference is to a guided missile, which, if its guidance system fails, will get out of control and follow a ballistic trajectory, i.e., a downward movement under the gravitational forces only.

This reference to the failure of a guided missile’s guidance system is clear in the text containing the earliest occurrence of the phrase to go ballistic that I have found. This text is The Face of War, an interview, by the Tribune editorial board, of Wayne Thomis, Tribune aviation editor, published in the Chicago Tribune (Chicago, Illinois) of Tuesday 19^th July 1966—in this interview, Wayne Thomis describes “how new techniques developed for the conflict in Viet Nam have given war new dimensions”:

Q. What are we running into in the way of anti-aircraft defense in North Viet Nam?
A. The Communists have […] the Russian SAM, a surface-to-air missile guided by radar both from the ground and by the guidance mechanism within the missile itself.
[…]
Q. How do we elude them?
A. In every attack formation there is a standoff watcher who gives warning when the SAM is seen to be fired and the area toward which it is directed becomes apparent. The formation breaks, perhaps making a violent turn, dropping altitude sharply, and changing direction. The SAM has a limiting factor in its ability to change course. If it exceeds these limits in trying to turn, it goes ballistic. Because it is fused, it generally explodes in the air.

An early figurative use of the adjective ballistic occurs in the following passage from Fortnight’s Anger (Manchester (England): Carcanet Press Limited, 1981), a novel by the English philosopher and author Roger Scruton (1944-2020):

He moved fragilely along the corridor and gallery, finding his way by instinct. On impulse he reached out as he passed the gulley of book-cases and, taking a small dusty volume from the shelves, slipped it into his pocket. The portraits followed him with their eyes. As he reached the stairs he began to run. He paused before the drawing-room door; Lord Gilroy poked his head from the library, waved cheerfully, and withdrew. Fortnight opened his mouth to speak, but since his voice seemed not to be obeying him, turned the handle and burst forward into the powdery air. A tall thin figure made willowy patterns against the window pane. Its hands were clutched behind it, the long, effeminate fingers clicking one against the other with a soft ivory noise. He turned and stared intently at Fortnight from black ballistic eyeballs.
‘You must be Kenneth Fortnight. I don’t believe we have met. My name is Adrian Holtius.’
And the doctor advanced, defining himself with delicate gestures of a hand which finally fluttered towards Fortnight’s lap and fondled the dead fingers that were hanging there.

These are the earliest figurative uses of the phrase to go ballistic that I have found, in chronological order:

1-: From Who Lost Lebanon? Why U.S. Policy Failed, by Lally Weymouth, published in the Los Angeles Times (Los Angeles, California) of Sunday 11^th March 1984:

In December, then-Israeli Defense Minister Ariel Sharon let leak to the press that he had negotiated a secret agreement between Israel and Lebanon. “When Habib found out about it he went ballistic,” Basil recalled. “He demanded President Gemayel not sign it.”

2-: From the radio programmes, published in the Liverpool Echo (Liverpool, Merseyside, England) of Thursday 12^th July 1984:

Four UK […]
[…] 4.40 Story Time: ‘Bootle Bounces Back’ by Richard Compton-Hall. Five stories: (4) ‘Bootle Goes Ballistic’.

3-: From TV’s hulky He-Man: Hero with a message ¹, by Jennifer Harper, published in The Washington Times (Washington, D.C.) of Wednesday 24^th October 1984:

Another traditional, kid-style dramatic device in the show is He-Man’s actual transformation from normal man to mean muthuh. Popeye had his spinach and Superman his phone booth. He-Man has a magic sword. When he holds it to the sky, gets struck by lightning a few times and thunders, “I have the power!,” it really brings the house down. The kids go ballistic, and the phrase echoes down the halls of even the most staid elementary schools.

¹ He-Man and the Masters of the Universe is a U.S. animated television series.

4-: From Reagan, Hill Set Budget Goal, by George C. Wilson, published in The Washington Post (Washington, D.C.) of Thursday 11^th July 1985:

An internal Defense Department memorandum recommends that the Army cancel its prized plan to build five light-infantry divisions to help offset some of a projected $250 billion congressional cut in defense budgets over the next five years, defense officials said yesterday.
[…]
Gen. John A. Wickham Jr., Army chief of staff, has championed the light divisions.
“Wickham went ballistic when he heard about the recommendation,” one Pentagon executive said in recalling the general’s reaction to the attack on his new design for the Army.

5-: From the column On Language, by William Safire ², published in several U.S. newspapers on Sunday 28^th July 1985—for example, in the Gainesville Sun (Gainesville, Florida):

According to a presidential speechwriter with a lifetime interest in crime-busting, the title Looney Tunes is in current use in law-enforcement circles as an adjectival phrase: “He’s Looney Tunes” is a policeman’s way of describing a subject as crazed, similar to the British “He’s bonkers” (from the sound imitative of a knock on the head) or the nautical “He’s only got one oar in the water” (which should properly be “He’s got only one oar in the water,” but there can be no arguing with idiom). Looney, of course, is from lunatic, meaning “moon-struck.”
A second hot detraction used at the highest levels of the American military is ballistic. Gen. John A. Wickham Jr., Army chief of staff, favors light-infantry divisions, but budget-cutters have removed all chances of the charge of an American light brigade. George C. Wilson in The Washington Post reported one Pentagon source’s description of the general’s reaction: “Wickham went ballistic when he heard about the recommendation.”
Apparently ballistic has replaced bananas in Pentagon use. The word is more apt than bananas, since it includes the connotation of a missile blazing skyward. In the synonymy of this special branch of derogation, bonkers is crazy, out of one’s mind, around the bend; bananas can be either bonkers or enraged to the point of flipping one’s lid; ballistic is driven up the wall in frustration, but not quite bananas.

² William Safire (William Lewis Safir – 1929-2009) was a U.S. author, columnist, journalist and presidential speechwriter.