The now well-known theory was used to explain a tiny discrepancy between Newtonian equations of Mercury's orbit versus observed data.
The year 2016 is the centennial of the second of Albert Einstein’s relativity papers. In 1905, he published his paper on what we now call Special Relativity, which accounted for the properties of bodies moving at constant velocity (along with four other landmark papers). With it, we came to understand time dilation, mass-energy equivalence, and many other non-intuitive phenomena. Some of these had been observed but were not explainable, and some had never been seen. His basic postulate that the speed of light was the same in all inertial frames, regardless of the motion of the source or observer, was a tough concept to grasp in a Newtonian world which had described so much and so well for so long.
Eleven years later, Einstein drastically increased the envelope of relativity’s reach with the General Theory which encompassed bodies which were accelerating, rather than restricted solely to constant velocity. This theory was far more difficult to prove than the special theory, for many reasons. Einstein himself used it to explain several inexplicable physics observations, to help make the case. See Die Grundlage der allgemeinen Relativitaetstheorie or The Foundation of the General Theory of Relativity.
Among the first was the aberration in the precession of the perihelion of Mercury, based on data spanning 400 years. In brief, the orbital plane of Mercury shifts slightly (precession) in accordance with Newtonian mechanics; the observations showed this was 5600 seconds of arc per century. Yet there was a difference between Newtonian theory prediction of 5557 seconds of arc per century and observed data of 5600 seconds of arc per century (only about 1% difference) at the extremes of the orbit—a tiny amount, indeed—but it was a still bothering astronomers and scientists. (You do have to admire the accuracy and precision of their collected data over the years; it was taken before our high-tech telescopes and electronic clocks.)
Even after accounting for all conceivable factors that might affect the orbit, such as the gravity of other planets as they orbited, the error could not be explained. There was even serious talk of an invisible planet named Vulcan that was causing this error (sorry, Mr. Spock).
What Einstein did was both significant and impressive: he used general relativity equations to calculate what the variation in the observed motion of Mercury would be, and showed that the entire discrepancy could be explained by it; thus, planet Vulcan was unneeded. A very readable and enjoyable recently book, The Hunt for Vulcan by Thomas Levenson, explores the subject in detail, there’s a well-written review of it in The Wall Street Journal, "Einstein, Destroyer of Worlds."
The technique of speculating about an as-yet unseen object is not unique to this situation: it was used to guide the search for unknown planets due to aberrations in the orbit of Uranus and led to the discovery Neptune. Similarly, the search for planet Pluto (now downgraded to a dwarf planet) was due to aberrations in the orbit of Neptune). More recently, the apparently successful hunt for the Higgs Boson at CERN to fill in a subatomic puzzle and equation.
Of course, engineers and scientists know that just explaining something which has been observed is valuable, but can lead to incorrect theories being accepted. After all, the Aristotelian Earth-centric model of the universe with its orbital epicycles worked quite well for explaining known celestial observations, and even predicting upcoming events such eclipses. Our present Copernican Solar-centric model only took over when there were many observations that could no longer be explained with the old theory. The "it explains it, so it must be correct" mindset can lead to false reasoning in many situations.
Einstein knew this, and knew that the strongest case for the validity of general relativity would be if it could predict something that had never been observed or even postulated. No one had imagined that a strong gravitational field would bend the fabric of space and time and so change the observed locations of stars, for example. The strongest early proof of general relativity was seen during of a total eclipse of the Sun in 1919, confirming Einstein's prediction that the observed positions of some stars would be shifted slightly as their light was bent as their rays passed the Sun (which cannot be seen during the day when the Sun is not eclipsed, of course). Most general relativity doubters were convinced by photos taken by teams which had traveled to a remote island off the west coast of Africa and to Brazil just for this test. Careful before-and-after examination of the photos showed the stars shifted exactly by the amount that Einstein anticipated.
[Note that to the average person, both special and general relativity are likely considered as "so what’s it to me?" theories. What they don't know is that GPS systems would be very inaccurate and good to no more than a few miles at best without the many general relativity-related adjustments that are made in its basic operation and computational algorithms, which are needed due to satellite and Earth accelerations.]
Have you ever worked on a project where a very plausible explanation for a problem turned out to be wrong (such as the invisible planet Vulcan was)? Have you ever used the prediction of unseen or unexpected phenomena as strong evidence that your explanation was correct?