This is one of two extremely tall flagpoles, relics from the 1939-40 World's Fair, standing beside the Pool of Industry. Atop each of the poles is an Art Deco eagle finial. It's long been rumored that the poles and the eagles came from Nazi Germany, which had at one point been planning to participate in the fair but later withdrew. According to an NY Times article from 1961 about the burying of a portion of the Flushing River for the upcoming 1964-65 World's Fair (the final article on this page):
The only [nearby] sign of the last World's Fair more than twenty years ago was a pair of gigantic flagpoles . . . They were topped by eagles of the Third Reich. The swastikas had been removed.The eagles are actually American eagles sculpted for the fair by Robert Foster, however. The apocryphal story about the Nazis is probably based on the similar appearance of the eagles and the Nazi German Reichsadler, although many American eagles of that era looked pretty much the same as well.
But why are the flagpoles so tall? Another NY Times article from the lead-up to the 1964-65 fair features the recollections of 1939-40 fair officials about the battle for elevational supremacy between the US and the Soviet Union. According to the officials, the Soviets constructed a tower at their pavilion that reached 188 feet in height, higher than the US flag flying atop the US Federal Building. In response, the Americans erected these two giant flagpoles — each of them taller than the Soviet tower — by the Lagoon of Nations (the Pool of Industry's predecessor). The Soviets then added a 75-foot-tall statue of a worker to the top of their tower. (Their request to put up a statue of Joseph Stalin was denied by fair officials, so they opted instead for the worker — "a figure that many American officials later swore was Stalin in his younger days".) Including the statue's 4-foot-tall base, the Soviet pavilion reached 267 (some sources say 259) feet into the air. So the Americans added a flagpole to the top of the Parachute Jump (yes, that Parachute Jump) and hung a flag on it, 270 (other accounts say 260 to 290) feet above the ground. And for good measure, they also increased the height of the two enormous flagpoles by the lagoon.
The part of that story about the flag on the Parachute Jump is well documented, as many Americans were apparently quite upset that the illuminated red star held aloft by the statue of the Soviet worker was higher than any American flag at the fair. It wasn't until Memorial Day of 1939, a month after the fair opened, that the flag was placed atop the Parachute Jump. (The Soviet worker statue, by the way, came to acquire a few nicknames from fair employees: "Big Joe" and "Joe the Worker", perhaps because of its glorification of an average Joe and/or because of its supposed resemblance to a young Joseph Stalin, and "The Bronx Express Straphanger", presumably because it kind of looked like a commuter hanging onto a strap on a busy subway car.)
The part of the story about the flagpoles by the lagoon seems a little questionable, however. As huge as these flagpoles are, I don't think they're over 188 feet tall. (120 feet maybe? 150 at most? It's hard to say just from looking at photos; I didn't think to try to estimate when I was there in person.) They could have been reduced in height after the fair ended, of course, but even in photos from the fair, the one positioned near the Soviet pavilion looks a bit shorter than the Soviet tower, although it's hard to tell for certain because the flagpole is closer to the camera. (Here's some footage of the other pole, located on the other side of the lagoon.) One 1939 Brooklyn Daily Eagle column states that, prior to the flag being hung on the Parachute Jump, the "red star is 259 feet, but no stars and stripes are even half so far up." The column is pretty smart-alecky, though, so I don't know if I should take its claims literally.
By the time the first season of the fair ended on October 31, 1939, World War II was under way and the Soviets had invaded eastern Poland. Shortly before their invasion of Finland on November 30 (both Poland and Finland had pavilions at the fair as well, located across the Court of Peace from the Soviets), they pulled out of the fair's 1940 season. Their pavilion was taken down and shipped back to Russia; an NY Times article about the dismantling, or "grim execution", of Big Joe was headlined "SOVIET'S WORKER AT FAIR IS 'PURGED' ". (Up to that point, adding insult to injury, Joe's red star had to be kept alight every night, even after the first season of the fair ended, because the height of the Soviet pavilion made it an aviation hazard.) You can see some cool photos of the pavilion and statue being taken apart here (use the navigation controls at the top of the right-hand column to get to the fourth page).
Fair officials made "strenuous efforts" to bring in the famed (and highly exploited) Dionne quintuplets as the "prime attraction" for the 1940 season, with a plan to build an exhibition space for the girls on the site of the Soviet pavilion, but they were unable to work out a deal. The Soviets were ultimately replaced by the American Common, an outdoor space intended to be used "for patriotic pageants, festivals and other spectacles". The common contained a bandshell and open-air theater, a "Village Green", and — wouldn't you know it — a towering flagpole that "dominate[d]" the site.
For the 1964-65 fair, the giant flagpoles from the lagoon were reused. They were moved to their current positions around the Pool of Industry, which was built on the site of the lagoon. In the photos I've seen, one flew an American flag and the other flew the flag of the fair. The poles are still used to display the Stars and Stripes at times, although they were flagless today.