Also by capturing one of them,.
What was known was that the thing could turn like a beast and climb like a Valkyrie. It was very maneuverable and fast. Nothing the US had could come close to it. However once the US captured one they found that it could turn great but when forced to roll it performed horrible. Also if you dove and made them chase you the engine of the Zero would stall out. With this knowledge strategies where adopted. Once the F6F came into the picture it was game over.
Also of note was that like with almost EVERYTHING in military terms the best pilots/soldier wins. The Japanese had pilots that where REALLY good and experienced that flew the Zero's. Once they started losing them they where not able to refill their ranks fast enough and keep the staffed long enough before the US killed them. This led to a feedback loop of great planes being staffed by worse and worse pilots while the US got better and better pilots.
Edit: Also to Add to Backgrinders comment Joe Foss a Marine Pilot ripped 4 Zero's apart in one pass to show you how vulnerable they where. Also I may be incorrect but Zeros did not have resealing gas tanks so one bullet = bad news.
It should be noted here that superior is a bit of an awkward word choice in your first question. The Zero was a fine aircraft, and provided superior performance in some ways, but it was inferior in others. The Zero was designed with extended range operational capacity first and foremost, so it was a very, very light aircraft. This contributed to making it particularly fast and agile, because speed is determined largely by power to weight, and a lighter mass means you can turn quicker with the same amount of force on the control surfaces.
The lighter weight also meant compromises though, primarily in defensive armor. Zeros were highly maneuverable and trouble one on one, but they were easy to knock down as well. Tough and durable is exceptionally important in combat, the Zero was neither of these things. Think of it like in American Football. It isn't much use being the fastest guy on the field if one hit knocks you out and you can't play the rest of the game.
The outbreak of war
/u/backgrinder gave an excellent look at the compromises in the Zero’s design. I would like to expand on that portion of the answer, as well as add some other details. The Zero was indeed designed for excellent range. The Japanese knew that being able to fly a long distance would be of great importance in any war they ended up in. As such, the Zero was built with the demand that it be able to fly a very long way. It was able to fly nearly two thousand miles without refueling. The Brewster F2F Buffalo and the Grumman F4F Wildcat—the Zero’s main competitors at the beginning of the war—were able to fly only about a third of that distance. To get this range, the main thing that had to be done was lose weight. This did indeed lead to excellent maneuverability.
So, why did the American planes not have this range and maneuverability? Design choices led to very different solutions to the same problems of naval aviation. The US Navy demanded rugged planes that could survive the rigors of hard landings onto pitching flight decks. To get reliable and rugged aircraft, you add things like more bracing, redundant systems, and heavier landing gear. Also, with the Wildcat features like pilot armor and self sealing fuel tanks were added. All of this added weight, which had costs of its own. As has already been mentioned, range and maneuverability were restricted. However, there were advantages as well. Stability and speed in a dive were improved. Also, the same rugged characteristics that allowed good peacetime reliability meant that more battle damage could be absorbed by a plane before it became crippled or destroyed.
Thatch weave
The ability to absorb damage is a good thing, but if that’s all you can do you’re still going to end up losing a fight. One on one, the Zero was able to get onto the tail of an American plane and destroy it. American pilots could not hope to climb or turn with a Zero, and individual brilliance was not enough to turn the tide. A new approach was needed.
Enter John Thatch of the United States Navy. He tested out a new system that bears his name, the Thatch weave. The wikipedia page has some good diagrams of how this innovation works. Instead of working one on one, American pilots flew as a team. Pair(s) of planes would weave through the sky. If enemy pilots attacked one element, the other would soon be on their tail. Now, the rugged nature of the Wildcat could be used to full effect. When they came under attack, they could absorb a few moments of fire while their teammate got into position to attack the pursuing plane. This tactic was employed to good effect at Midway, and it was soon picked up by the Cactus Air Force on Guadalcanal. The Thatch weave allowed for much greater success against the Zero. Since it had no pilot armor or self-sealing fuel tanks, the Japanese planes could not simply absorb battle damage and would quickly be destroyed. However, if the weave was not executed perfectly US pilots were quite vulnerable. Any mistake would leave US pilots very much alone in a plane that could not hope to match the acrobatics of their opponent.
Attrition and training
One sometimes overlooked element of the Pacific theater was attrition. The Japanese entered the war with a cadre of highly experienced pilots. These men had been in combat for years before Pearl Harbor, and had learned a number of lessons that only combat can teach. In their incredible Zero fighters, they were able to fly circles around less experienced opponents. Those same Zeros would catch fire or succumb to damage quite easily, and unless the pilot was over friendlies he had very little chance of rejoining the war. Further, the lack of pilot armor meant that these highly experienced pilots were vulnerable to a single stray bullet that could end their career—or their life. There were dramatic losses of experienced pilots such as the Battle of Midway, and there was also slow attrition in the skies surrounding Guadalcanal and New Guinea. On the other side of the coin, American pilots received relatively lavish training and their more survivable aircraft allowed men to return home when a Zero would not have been able to.
Planes at the end of the war
The development of the Hellcat was detailed by /u/pmaj82. The development of succeeding waves of fighters is the final part of how the Zero was countered by the US. The F6F Hellcat could match or beat the Zero in every aspect except range. It was rugged and survivable like the Wildcat, but with greatly added speed, rate of climb, and agility. And the Hellcat wasn’t alone. It was joined by the Chance Vaught F4U Corsair. (I like just saying that phrase. Chance Vaught F4U Corsair, try it!) These planes were as iconic to the Pacific theater as the P-51Mustang and P-47 Thunderbolt were in the European theater. In addition, there were Mustangs deployed in the Pacific as well, primarily due to their incredible range and high altitude capability making them ideal escorts for the B-29 Superfortress.
Now, the technical capabilities of these planes is somewhat obvious. The less obvious event that went hand in hand with their success was the production of these fighters. American aircraft production dwarfed that of the Japanese. The American economy was able to produce huge numbers of highly advanced aircraft, and was agile enough to switch production from prewar designs to those designed during the war. While most of the other combatants in WWII were able to upgrade existing airframes during the war with more powerful engines and improved subsystems, the US was able to field entirely new fighters with seeming ease. Then, the US was able to make these new planes in the thousands. Paired with giving their pilots sufficient training, this prodigious rate of production resulted in a highly capable force that was able to island hop all the way to the doorstep of the Japanese home islands. Along the way, the Zero was first supreme, then contested, and finally driven from the skies by a succession of planes, tactics, and pilots.