What does a film's "bowing" mean?

To 'bow in' at one's debut

Use of the verb bow in the sense of "debut" goes back at least to the 1920s, although it more often takes the form of the phrase "bow in" when used in this way. For example, from "Belasco Season On Next Sunday," in the Washington [D.C.] Times (August 20, 1922):

The regular 1922–23 dramatic season will bow in for Washington next Sunday night, August 27, when the Belasco Theater throws open its doors for the first of the fresh output of drama prepared for the playgoer.

And from "At the Theatres," in the Sweetwater [Texas] Reporter (September 6, 1938):

"There's Always a Woman," romantic comedy co-starring Joan Blondell and Melvin Douglas, will bow in at the Texas theatre today. Based on a popular magazine story by Wilson Collison and adapted to the screen by Gladys Lehman, the story of "There's Always a Woman" tells of the zany efforts of Joan Blondell to out-sleuth her sleuthing husband, Melvyn Douglas.

And from "Midget Autos to Race Sunday: Fresno Season Will Open at Speedway," in the Madera [California] Tribune (May 2, 1940):

The 1910 midget auto racing season will bow in at Fresno Sunday when speed aces from all sections of California will vie for fame and money on the Airport speedway track.

From Chuck Moore, "Keeping Posted," in the San Antonio [Texas] Register (October 17, 1947):

In the event you were not aware of it, the King Cole trio will lie the first sepia act to play at Frank Dailey's Meadowbrook nitery in Newark, New Jersey. Trio will bow in along with the Tony Pastor ork on October 28. They will play a Detroit nitery engagement the following week and then continue with their concert tour ...

From "Wet Year Is Seen at Diaper Meeting," in the Breckenridge [Texas] American (May 4, 1955):

NEW YORK (U.P.)—Diaper service operators, assembled for the 10th annual convention of their infant industry, had good news from their vital statistics department Wednesday—it looks like 1955 will be a very wet year.

The birth rate is the only statistic vital to diaper laundrymen, and prospects are that nearly 4,100,000 babies will bow in as U.S. citizens and potential customers this year. That's almost 100,000 more than 1954.

From Jack Hand, "Dodgers, Yankees Begin Title Defense" in the [Urbana, Illinois] Daily Illini (April 14, 1964):

Pittsburgh expects improvement from the 1963 cast, particularly Bob Bailey and Donn Clendenon. The "second" openers will stretch from April 15 to April 22 when St . Louis and Milwaukee finally will bow in at home.

And from "Football Exhibition Slate Opens Today" in the Columbia [Missouri] Missourian (August 8, 1975):

Six new coaches will bow in during the preseason preseason season play over the weekend On Saturday Bart Starr's Green Bay Packers meet Buffalo at home, Ted Marchibroda's Baltimore Colts play at Denver, Jack Pardee's Chicago Bears travel to San Diego, the Houston Oilers under Bum Phillips are at New Orleans and the Kansas City Chiefs with Paul Wiggin at the helm entertain the Cardinals.

'Bow' in place of 'bow in'

But bow in a very similar figurative sense meaning "to politely acknowledge an audience" occurs even earlier, in an advertisement for Wanamaker's department store in the Philadelphia [Pennsylvania] Ledger (August 10, 1918):

Not one frock has been shown before, as they have only just arrived, and Monday morning they will bow in all their fresh fine newness. They were bought for considerably less than usual, making possible the purchase of two delightful frocks for the usual price of one.

Also, from Chuck Moore, "Keeping Posted," in the San Antonio [Texas] Register (January 6, 1950):

Another first coming up will be the all-colored stage version of "Tobacco Road," with Powell Lindsay, founder of the Negro Drama group, in the role of Jeeter Lester. Show will bow at the Howard theatre in Washington for two weeks beginning January 31 ....

This example is from the same syndicated writer who used "bow in" to mean "debut" in an example from October 17, 1947 (above). Here the meaning of "bow" is a bit different from its meaning in the other examples cited in this answer: it seems to be equivalent to "take bows [after performances]" rather than "bow upon being introduced."

A headline in the [Abilene, Texas] Optimist (November 29, 1958):

'Wife' Draws Critic's Raves, Will Bow at ACC Thursday

From "Column Starts on Sixth Year," in the [Washington, D.C.] American University Eagle (September 23, 1959):

Besides these recent activities of bringing out a new book and a new television series, [Max] Shulman will bow on TV as an announcer. He'll deliver the Marlboro commercials on the "Dobie Gillis" telecast. He also penned the lyrics to the show's theme song.

From "Summer Series Makes Its Bow," in the Orange [Texas] Leader (June 18, 1961):

On a visit with the Crosby brothers—Lindsay, Dennis and Philip—host Charles Collingwood (left) drops in at Lindsay's home to greet the youngest brother's wife, Barbara, and their son; David, in the first program of the "Person to Person" summer series. The show will bow on CBS-TV and Channel 6 next Friday at 9:30 p.m.

From "Play Produces 'Electric' Effect," in the [University Park, Pennsylvania] Daily Collegian (May 28, 1964):

After months of hardship and conflict "Emperor Jones" will bow at 8:30 tonight in Schwab.

Without seeming to go overboard, there is only one word to describe "Emperor Jones"—ELECTRIC.

From "Car 'Biggies' Join the Switch," in the [Palm Springs, California] Desert Sun (April 5, 1977):

The [Lincoln] Versailles will bow in mid-April, about the same time as Chrysler introduces two new car lines, the Dodge Diplomat and Chrysler LeBaron.

From the Bob Holmes, "On Media," in the [Santa Monica, California] Corsair (February 21, 1980):

"Consumer Crusaders" starring David "Action 4" Horowitz, will bow in September. "Crusaders" will profile individuals and organizations that "fight back," according to an NBC press release. Horowitz will continue to occasionally test various products.

From Bob Thomas, "It's Laughter in the Aisles," in the Canberra Times (January 29, 1987):

The long-awaited and once-postponed Ishtar will bow in May. That's Elaine May's movie with Warren Beatty and Dustin Hoffman as a couple of hapless singer-songwriters who finally find a gig in Morocco.

From Linda Dailey, "Play Explores Racial Stereotypes," in the [Abilene, Texas] War Whoop (April 22, 1983):

McMurry's first production with a primarily black cast will bow on stage April 22nd in Ryan Little Theatre.

"Purlie Victorious" written by black actor Ossie Davis, is the play and directing it will be Marion Castleberry.

And from "Rejoice! Is Resurrected," the Indianapolis [Indiana] Recorder (June 30, 2000):

The first release from [the revived gospel record label] Rejoice! will bow in the fourth quarter of 2000.

Conclusion

Recorded instances of bow used in the sense of "debut" or "arrive" go back almost a hundred years, most often in the form "bow in" (which is the mirror opposite of "bow out," meaning "withdraw" or "depart"). The underlying notion here seems to come from the practice of entertainers' bowing to the audience as they come on stage and are introduced—although that practice in turn may derive from the centuries-older practice of bowing and curtsying during introductions in polite society.

Variety is a leading entertainment industry trade publication which launched in 1905, making it one of the oldest trade papers still around. According to an article on the topic published by Variety itself, from the start it purposefully used its own slang "in part [as] a device to fit long words into small headlines, but... also to create a clubby feel among the paper’s entertainment industry readers." To the extent that other entertainment industry publications want to appear to be "inside" the business rather than aimed at the general public, they tend to adopt Variety's slang.

Bow is one of those industry slang words. Though probably derived from the practice of stage entertainers bowing to the audience at the beginning of their performance, exactly where it came from is uncertain. However, since Variety now helpfully publishes a dictionary of their own slang, we can at least be clear on what it means in this context:

bow — (n.) opening or premiere; (v.) to debut a production; 
“The pic’s bow was in January”; 
“The Nederlander Organization will bow its revival of ‘Wonderful Town’ next year.”

Just for fun, let me mention probably the oddest and most iconic Variety slang word, ankle, meaning to quit or be dismissed from a job, without necessarily specifying which. It is meant to suggest walking, as in walking away from a job.