The person who came up up with the theory of continental drift was Alfred Wegener. He published his theory in 1912.
One of the issues with the theory that geologists at the time had, was that Wegener was not a geologist, but a meteorologist. He was publishing a theory that wasn't associated with his field of science. The other issue the geologists had was based on the commonly held opinion:
that the oceanic crust was too firm for the continents to "simply plough through".
When he published his theory, Wegener did not propose a means by which the different land masses could break away from each other.
Initially, some geologists could only conceive the idea that ocean waves might be responsible for breaking up land masses, but they couldn't reconcile the fact that the lack of sediments and the clean breaks in land masses would not support Wegener's theory. Without knowing about plate tectonics the theory of continental drift was difficult to support.
Another reason why Wegener's idea was not initially accepted was because of the way he proposed that continents used to fit together. This was because of the assumption most people had was that the continents split along the lines of coast lines and not the 200 m isobath proposed by Wegener.
Wegener came up with the idea of continental drift by noticing that all the major land masses appeared to fit together like a jigsaw puzzle.
It wasn't until the early 1950s when data from paleomagnetic studies of India showed that India had once been in the southern hemisphere that data started to support Wegener's theory. Also, it wasn't until the 1960s that sea floor spreading data was available to support the theory.
One could speculate on controversy due to people ferreting around in their own area of expertise and developing explanations that are at odds with data from other areas but much is due to the development of extensive data sets to show the big picture. But certainly people have a tendency to be personally vested in their own theories and that, too, is needed to demonstrate the robustness of new ideas.
I'm not sure the discovery is so important 'in its time' - it made its own importance and would have been so whenever it was developed. It did have importance in providing models for resource exploration to meet the rapidly increasing demands of the time. It also provided tangible benefit from the large government and private investment in science and technology of the period.
Wegener's theory was based partially on the shapes of the eastern coasts of the Americas versus the shapes of the western coasts of Europe and Africa. Wegener was hardly the first to notice this similarity. What Wegener added was fossil evidence; there appeared to be something much deeper than a mere random similarity of coastlines. What Wegener didn't add was a mechanism.
What he did add was exactly what geologists had been fighting for about a hundred years, which was religion. Nineteenth century geology was a battle between those who believed in the Bible (literally) and those who looked at the geological evidence. Early nineteenth century geologists explained the diversity of the Earth's geology via catastrophism, with Noah's flood playing a predominant role. Later geologists looked at the evidence and saw no signs of catastrophes. They instead adopted a theory of uniformitarianism.
One big problem was the 1920s Scopes trial in the US. Scientists in the US were hard pressed at the time. One counter reaction amongst the American scientific community to the Scope trial was a rejection of anything and everything that remotely hinted of religious catastrophism. This included Wegener's continental drift.
Plate tectonics is not Wegener's continental drift. Unlike Wegener's continental drift, plate tectonics has a mechanism. Plate tectonics is founded on a lot more than a mere similarity of continental outlines. A mountain of evidence had accumulated in the time that intervened between Wegoner's unfounded claims and J. Tuzo Wilson's 1963 article that proposed plate tectonics.
The evidence that turned plate tectonics into the accepted science in five short years included
The distinction has already been made, above, between Wegener's 1920s Continental Drift (he actually called it Continental Displacement, the more whimsical term was created later by others to disparage Wegener's idea) and Plate Tectonics which was developed 50 years ago, during the mid-1960s. It took several years for the plate tectonics model to be accepted, but as far as 'revolutionary' scientific theories are concerned, it actually caught on relatively quickly. Marie Tharp discovered mid-ocean rifts (1956); Harry Hess explained how they work and invoked the idea of subduction zones (1961); Morley, Vine, and Matthews proved seafloor spreading by using paleomagnetism (1963); Tuzo Wilson proposed hot spots and transform faults (1963); Isacks, Oliver, and Sykes mapped a subduction zone with seismic (1967); Jason Morgan divided the planet into plates (1968). Of course many others were also involved, but by 1968 only a few stubborn holdouts opposed plate tectonics theory.
To the point of the original question: It took massive paleomagnetic, seismic, and thermal readings to convince most geologists that the crust moves laterally. It took a while before technology reached a level to prove tectonics. Further – and this is important – scientific theories need to be testable and measurable. With advanced technology, it became possible to predict plate movement and with accurate measurement (especially GPS), it finally became indisputable that the crust is moving in the ways the theory predicted.