This is a difficult question to answer, but as someone in the sciences, I can provide a semi general answer and then say what I know about early flight.

First of all, when we look back on these great advances in technology, we generally see them as "giant leaps" instead of the small steps they actually were.

Einstein, for example, while he did take a giant leap with his General Relativity, the things that actually made him famous (special relativity, photoelectric effect) were not giant leaps, but were really just taking the work of previous scientists and expanding on it. Special relativity, for example, was just expanding on the Lorentz equations that had been published decades before. A part of one of my undergraduate physics classes was my professor putting an emphasis on how 99% of advances in the sciences (which lead to advances in engineering/technology) were just great men building onto the works of other great men.

Okay, back to aviation...

The Wright Brothers are generally considered by the majority of historians to be the first to fly in a motorized airplane. They were not alone in the race to be first, however, and there were other people working on similar flying machines all around the world.

Centuries before their time, gliders were already being experimented with and the general concept of lift was known. In the early to mid 19th century, George Cayley was working on heavier than air fixed wing aircrafts and was a huge inspiration to the Wright Brothers, with Wilbur saying the following:

"About 100 years ago, an Englishman, Sir George Cayley, carried the science of flight to a point which it had never reached before and which it scarcely reached again during the last century."

— Wilbur Wright, 1909

When speaking of the advancements in flight, I believe you are missing the mark by placing the most amazing advancements as those that happened from ~1900 to ~1960, when really the biggest advancements seemed to be during the 19th century. Remember, its easier to add on to someone else's work than it is to be the one to originally come up with the idea, and while we shouldn't discount the amazing things the Wright Brothers did for aviation, the work Cayley did was almost certainly more important.

I return to Einstein for an example... Without Einstein, someone surely would have written a paper on Special Relativity and the Photoelectric Effect sometime in the early 20th century. But, the more abstract and advanced concept of General Relativity might have gone undiscovered for decades. THAT is what makes Einstein special.

In the same way, had the Wright Brothers decided to pursue automobiles rather than aircraft, someone else would have flown a fixed wing airplane within a few years of the first flight at Kittyhawk. Had it not been for Cayley's work, flight could have been pushed back decades.

The other thing I believe a lot of people neglect is that the technology that greatly advanced flight was technology being invented for other uses. The engine was really the driving (sorry) force behind the advancement of aircrafts, but the initial development of engine technology was not done with aircrafts in mind.

So now I've described how flight was actually something being worked on all over the world by a variety of different people. We see a lot of other quick advances over the first few decades of the 20th century, and they all seem keep up with the advancement of flight (or the other way around, if you'd like.) Automobiles advanced at a very fast rate, and if you look at the top speed of cars over time, you'll see it didn't take long for people to go fast. By 1904, the land speed record for a combustion engine after traveling a mile was 91mph (Ford 999 Racer).

The technology had finally caught up, as that was the main thing holding flight back, and so flight, well, took off.

And then... World War 1.

World War 1 was fascinating and horrifying because it gave a bunch of countries the opportunity to test all the new technology that had been developed over a short amount of time. If war is good for anything, its good for pushing technological advancements.

The aircraft was immediately useful for reconnaissance, and eventually was used to drop bombs and shoot down enemy planes. The need for faster, more manueverable, and higher flying aircraft went through the roof, and countries all over the world started competing for the best aircraft.

As we continue on to 1960, we see an arms race all over the world and the increasing importance of aircraft on the battlefield. By WW2, the most important piece in a Navy was not the Battleship but the Carrier, specifically due to the use of aircraft. Countries all over the world were constantly trying to outdo the competition, and technology advanced as expected.

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Essentially, to summarize, the aviation tech tree became much easier to unlock following 1903 because that was when combustion engines really started to... pick up steam (sorry.) The concepts of flight were well known by 1903, all that was missing was a good power/weight ratio engine to generate the thrust required to get lift. Once engine tech advanced, people like the Wright Brothers were able to use the theory from previous great minds and brand new engine tech to finally take a controlled flight.

When you see these giant leaps, you have to remember that you're looking at things in the past and you're not necessarily seeing all the steps taken along the way. The path is far more linear than you'd expect.

Gliders had been known for centuries. The major breakthroughs that were needed for powered flight were 1) a liquid fuel with high energy density, 2) precisely machined engines, 3) development of the mathematics of airflow and lift. With these elements in place the "powered glider" was the next step, and from there, trial and error led to the development of flight control systems (which is what really set the Wright Bros apart from their competition).

Rockets are an entirely different story that are only sort of related to the history of flight. Like gliders, rockets had been around for centuries but they were essentially self-propelled shells. The analogy, therefore, between gunpowder cannons and flight was not lost on 19th century author Jules Verne who wrote De la terre à la lune, an early science fiction novel, in 1865. His novel depicts a gigantic underground cannon, with its muzzle at ground level, being loaded with guncotton and used to fire a livingroom-sized aluminum capsule at the moon with astronauts and a dog aboard. The British author H.G. Wells also wrote a novel about exploring the moon in 1901. Rocketry, as a science, has the strange distinction of being strongly and directly inspired by fiction. Verne's book was read by the Russian Konstantin Tsiolkovsky, while HG Wells was read by Robert Goddard. Starting in the early 1900s, these two pioneers developed both the theory and practice of liquid fuel rocketry.

The main barrier between primitive shell-type rockets and true rocketry is 1) rocket-equation efficiency 2) onboard guidance. To boil down Tsiolkovsky and Goddard's contributions: Tsiolkovsky developed the math of how a rocket's acceleration climbs higher and higher over time, because the rocket's mass declines as it burns fuel. The rocket equation gives you a number called delta-V which is how fast the rocket is going (from the perspective of a ground observer) when it runs out of fuel. Delta-V's for orbiting earth or reaching the Moon are easily calculable constants. Tsiolkovsky realized a rocket would be more efficient if it could continually shed weight, and popularized the idea of rockets with discardable stages. The development of liquid fuel engines from aviation was useful to the first rocket pioneers, because it had high energy potential per unit mass. At the other end of the rocket equation was the idea of impulse, which basically means that a rocket only goes fast if the stuff leaving it out the rear end goes really, really fast; the noise and heat it makes is superfluous. Previous rockets were slow, controlled explosions, using black powder, that inefficiently shoot out lots of hot high pressure gas. The Swede Gustaf de Laval had done groundbreaking mathematical and practical work on a nozzle design (for steam turbines, not rockets) that actually transformed hot, high-pressure, fast moving gas into relatively cool, low-pressure gas that was moving even faster - faster than the speed of sound! Goddard realized the value of this work and borrowed it for his experimental rockets. By unleashing more of the potential impulse in each unit of fuel, the de Laval nozzle greatly increased the Delta-V achievable even with primitive rockets. From there, it was just a matter of gaining more and more working experience as the technology scaled up. Unfortunately for Goddard, the work of Tsiolkovsky (and his German contemporary Hermann Oberth, mentor to Wernher von Braun) fell on more fertile ground in Europe than America, where interest in rockets stalled. By WW2, the most scientific and military interest in rockets was in Russia and Nazi Germany. The Germans developed a hybrid military-scientific program that made genuine scientific advancements while also developing guided ballistic missiles as "vengeance weapons" against Allied civilian populations. The weapons didn't achieve much militarily, but they had increasingly sophistcated on-board controls, culminating with the V2 rocket. Even during the war, the German engineers working on these weapons were aware of the applications to space. They even launched one rocket straight up, achieving a top altitude of 176 km. After the war, many of the German personnel were deliberately seized by the American military in Operation Paperclip, which marked the unofficial beginning of the Space Race, which sharply intensified four years later when the USSR developed atomic weapons. While the Soviets had many firsts in the fields of low-Earth orbit and low-mass (robotic) exploration of other planets, it was the mind-bogglingly huge Saturn V rocket program, led by von Braun, that built the first rocket powerful enough to send a manned capsule to the Moon.

The short answer to your question is: advancements in propulsion. As u/Picklesadog mentioned, there were advancements made during and because of two world wars.

But being able to mate a sufficiently powerful engine with an airframe as what transformed flight into something useful.

Early 20th century advancements in naturally aspirated internal combustion engines really enabled more complex aircraft to take flight. Leading into WWII, supercharging them allowed aircraft to fly higher, in thinner air.

Post WWII, the development of turbojet engines was another quantum leap forward in aircraft complexity.

The advancement hasn't let up though. Consider that a single turbofan engine of the current model 747 produces as much thrust as ALL EIGHT turbojet engines on a B-52.

Another thing to consider in modern times is efficiency. We've unlocked power, but it's not the be-all and end all. There will never be another Concorde with current technology. There's just no appetite anymore for speed at all costs. And the laws of physics (drag increases with the square of the speed of an aircraft) places limits on realistic cruising speeds.

Technically speaking, there is a big difference between heavier than air flight and rocketry/spaceflight. Airplane design is governed by the four major forces of lift, thrust, drag, and weight. Airplane design was largely held back by a lack of powerful engines and lightweight structural components. Rocketry, on the other hand, relies on accelerating a payload to a very high velocity. Once it is in space, it doesn't need to be very rugged or aerodynamic, but it requires a large force and a lot of energy to accelerate even a small payload to orbital velocity. Rocketry was largely held back by engine design and guidance systems. There were major developments in both areas no doubt between 1903 and 1969, but to put them in the same timeline is a little misleading.

The biggest boon to aircraft designs was a result of both of the world wars. WW1 saw experimentation with aircraft as spotters and fighters, with limited use as bombers and ground attack. However, post WW1 many nations began to develop airforces at an accelerated rate. The money from government contracts fed the first aviation companies, like Boeing, Junkers, Mitsubishi, etc. WW2 again saw further investments in aviation technology, including the development of long range high altitude bombers and jet propulsion, among dozens of other quality of life improvements to aviation. Post-war contracts further pushed experimentation with various projects like flying wings, ultra-high altitude aircraft, and supersonic flight. By 1969, we had achieved many massive aviation milestones, via the Blackbird, Boeing 747, Concorde, and nearly the F-14 Tomcat.

Rocketry really began to come into being in the early 1920s with Robert Goddard experimenting with liquid fueled rocket engines in 1923. Gunpowder based rockets had seen use since antiquity, but had only limited usage for fireworks and some military applications. The development of liquid fueled rockets was the important step in rocketry because it allowed extra control in the ascent of rockets, as well as denser energy storage. The next great leap of Rocketry was mainly at the hands of Wernher von Braun, first working in Nazi Germany with the V2 rocket program. This program made important breakthroughs in ICBM control designs, launching rockets that escaped earth's atmosphere and could accurately hit targets at extreme distances. Post WWII, Von Braun was put to work by the Americans in our own rocket programs. Other German scientists had been swooned by the Soviets in their own rocket programs, with both nations racing to develop ICBM technologies. Since this was the beginning of the Cold War, and nuclear ICBM's were a top priority, there was essentially unlimited government funding on both sides to get the technology operational. Even after we had developed the technology for ICBM's, we got caught up in the Space Race after the launch of Sputnik in 1957. We simply kept building bigger and bigger rockets. Rocket science isn't really all that difficult. Life support systems in space follows very similar rules to life support on a submarine, after all. The only hurdles left after 1957 was to simply build a big enough rocket engine that could carry men and their equipment to the moon and back.

Simply put, rocketry and aviation took a central focus from the military between 1903 and 1969. We were able to achieve the massive leaps in innovation through massive investments, two world wars, a cold war, and a dozen or so exceptional designers.