Before we could look for high energy photons in the sky, we had to know that such photons existed.
On November 8, 1895, Wilhelm Conrad Röntgen discovered X-rays by accident. "[H]e found that, if the discharge tube is enclosed in a sealed, thick black carton to exclude all light, and if he worked in a dark room, a paper plate covered on one side with barium platinocyanide placed in the path of the rays became fluorescent even when it was as far as two metres from the discharge tube." (From a biography of Röntgen on the Nobel web pages) The fluorescence was occurring too far from the discharge tube to be caused by the cathode rays from the tube. From this and subsequent experiments, he inferred that a very energetic form of radiation was responsible, which he named X-rays. He did not know at the time that these rays were in fact a form of light, this was later discovered by Max von Laue and his pupils in 1912. (For another account of the story, see this page.)
In 1900, Paul Villard discovered gamma rays, which were thought to be a much more energetic form of X-rays (well, they are, but at that time X-rays had not been shown to be a form of light). It was not until 1914 that Earnest Rutherford showed that gamma rays were electromagnetic in nature (i.e. photons) with wavelengths shorter than X-rays.
Once we knew that X-rays and gamma rays were a form of light, we had to make a mental jump to look for them in the sky.
Between 1912 and 1914, Hess and Kolhörster made balloon flights during which they detected extraterrestrial "radiation" (they actually detected cosmic rays, which are not radiation, but charged particles moving very near the speed of light). The discovery of particles from outer space hitting our atmosphere began the study of cosmic rays, but also may have inspired other scientists to look into the heavens for sources of other kinds of radiation.
In 1933, Karl Jansky was investigating the use of short-wave radio for transatlantic telephones. He erected an radio antenna and discovered three types of static. One was from nearby thunderstorms and another from far-away thunderstorms. The third type was a steady faint "hiss" which rose and fell slightly on a 23 hour 56 minute cycle. This cycle is what one would expect from a source from the "fixed stars". It turned out that this third source of static was actually radio emission from the Milky Way Galaxy! Thus beginning the study of radio astronomy. (For the full story see this page at the National Radio Astronomy Observatory site.)
What does the beginning of radio astronomy have to do with high energy astronomy? Well, this was the first time anyone had shown that radiation of any energy could be detected from the galaxy - outside our Earth. In fact, this could be considered the beginning of multiwavelength astronomy!
The last step to doing high energy astronomy is to get detectors above the atmosphere.
On March 16, 1926, Robert Goddard made the first successful flight of a liquid propellant rocket, sending it to a 41-foot altitude. This was a necessary step to sending rockets even higher. (Time has a biography of him, naming him one of Time's 100 Scientists & Thinkers.)
Twenty-three years later, in 1947, the Naval Research Laboratory sent up an instrument on a V2 rocket which detected X-rays from the Sun in a 5-minute observation. This was the first time X-rays were observed from non terrestrial source.
Finally, the last hurdle to doing long observations of X-rays and gamma rays is to be able to have a detector orbit the Earth. On October 4, 1957, the successful launch of Sputnik-1 by the USSR was the first step to conquering that hurdle.