As I was reading about the heliocentric model, a question came up: How was Nicolaus Copernicus able to figure out that the sun is at the center of the solar system, and that all planets orbit around it, even though the telescope had not been invented yet? During his time the geocentric model was the standard.
Copernicus did not figure out that the Sun was at the centre of the solar system. He merely proposed that a heliocentric model simplified the calculation of planetary orbits. Epicycles (circles within circles) were still needed in the Copernican model, for instance for the orbit of Mars, but there were fewer of them than in the Ptolemaic model.
The Copernican model received a boost when Kepler, discovering that planetary orbits were elliptical, removed the need for epicycles completely, thus increasing Copernicus' advantage over Ptolemy.
The first evidence that the Copernican model was a true representation of actual reality came with the observation of stellar parallax in 1806. Until then, it was preferred only because it was a simpler description - a much simpler one, after Kepler.
Basically, because it was a more elegant solution to describe the motion of the planets. This could be see easily without any telescopes. The big difference was that it needed way less so called epicycles (he still needed those as all his orbits were circular not elliptic):
One of the glaring mathematical problems with this model was that the planets, on occasion, would travel backward across the sky over several nights of observation. Astronomers called this retrograde motion. To account for it, the current model, based on the Greek astronomer and mathematician Ptolemy's view, incorporated a number of circles within circles — epicycles — inside of a planet's path. Some planets required as many as seven circles, creating a cumbersome model many felt was too complicated to have naturally occurred.
Besides that, the idea wasn't entirely new. Aristarchus proposed the idea of a moving earth in 250 BC and Copernicus was probably familiar with that. (Source)
As noted in @SpaceBread's answer, Copernicus did not 'discover' heliocentricity (there were no observations to support his theory) nor did he invent it (that was done two thousand years earlier). Nor was his his model better -- because he insisted on circular orbits it was contradicted by observations and a worse fit even than Ptolemy's model.
What he did was popularize the heliocentric model and argue that its simplicity made it preferable even though it was clearly wrong.
The real breakthrough was Kepler's who showed that a heliocentric model with elliptical orbits gave an excellent fit to observations. And then Newton showed why Kepler's model worked.